SlideShare a Scribd company logo
A Brief Introduction to Statistics

What is Statistics
What is Statistics?
1. The science that deals with the collection,
organization, presentation, analysis, and
interpretation of numerical data to obtain
useful and meaningful information
2. A collection of quantitative data pertaining
to a subject or group. Examples are blood
pressure statistics etc.
A Brief Introduction to Statistics

Branches of Statistics
Two branches of statistics:
1. Descriptive Statistics:
Describes the characteristics of a product or

process using information collected on it.
2. Inferential Statistics (Inductive):
Draws conclusions on unknown process

parameters based on information contained in
a sample.
Uses probability
A Brief Introduction to Statistics

Data
DATA is any quantitative or qualitative
information.
Types of Data:
1.

Quantitative – numerical information obtained
from counting or measuring (e.g. age, qtr.
exam scores, height)

2.

Qualitative – descriptive attributes that cannot
be subjected to mathematical operations (e.g.
gender, religion, citizenship)
Measures of Central Tendency and Dispersion

The Measures of Central
Tendency and Dispersion
Statistics use numerical values used to
summarize and compare sets of data.
 Measure of Central Tendency:
number used to represent the center or
middle set of a set of data
 Measure of Dispersion (or
Variability): refers to the spread of
values about the mean.


(i.e., how spread out the values are with respect to the mean)
The Measures of
Central Tendency
Measures of Central Tendency and Dispersion

Measures of Central
Tendency

The Measure of Central Tendency:
1. Mean - the (arithmetic) average (or
the sum of the quantities divided by the number of
quantities)

Median – the middle value of a set
of ordered data
3. Mode – number in a data set that
occurs most frequently
2.
Measures of Central Tendency and Dispersion

The Mean
It‘s known as the typical ―average.‖
 It is the most common measure of central
tendency.
 Symbolized as:
◦ x for the mean of a sample
◦ μ (Greek letter mu) for the mean of a
population
• It‘s equal to the sum of the quantities in the
data set divided by the number of quantities


x

x
n
Measures of Central Tendency and Dispersion

The Mean
Example 1
Find the mean of the numbers in the
following data sets:
a.

b.

3, 5, 10, 4, 3

x

3 5 10 4 3
5

85, 87, 89, 90, 91, 98 x

540
6

90

25
5

5
Measures of Central Tendency and Dispersion

The Mean
Example 2
The table on the right
shows the age of 13
applicants for a job in a
factory in EPZA. What is
the average age of the
applicants?
(Adapted from DOLE-BLES i-Learnstat
module on Measures of Central Tendency)

Solution: x

318
13

24.5
Measures of Central Tendency and Dispersion

The Weighted Mean
It is a mean where some values contribute
more than others.
 Each quantity is assigned a corresponding
WEIGHT


(e.g. frequency or number, units, per cent)


The weighted mean is equal to the sum of the
products of the quantities (x) and their
corresponding weights (w), divided by the sum
of the weights.

x

wx
w
Measures of Central Tendency and Dispersion

The Weighted Mean
Example 3
SCORE

NO. OF
STUDENTS

5

8

4

6

3

3

2

2

1

1

The table shows the scores
of 20 students in a 5-item
Math IV seatwork.
Find the average score of
the class.
Measures of Central Tendency and Dispersion

The Weighted Mean
Example 3 SolutionSCORE

x

78
20

3.9

PRODUCT

5

8

40

4

6

24

3

3

9

2

2

4

1

Multiply the scores by
the number of
students, then find the
sum. Finally, divide by
the total number of
students
The average score is

NO. OF
STUDENTS

1

1

sums

20

78
Measures of Central Tendency and Dispersion

The Median
Used to find the middle value (center) of a
distribution.
 Used when one must determine whether
the data values fall into either the upper
50% or lower 50% of a distribution.
 Used when one needs to report the
typical value of a data set, ignoring the
outliers (few extreme values in a data
set).


◦ Example: median salary, median home prices in a market
Measures of Central Tendency and Dispersion

The Median
How to find the median:



Order the data in increasing order.
If the number of data is ODD, the
median is the middle number.

If n is odd, the middle number in n observations is the
(n + 1)/2 th observation



If the number of data is EVEN, the
median is the mean of the two middle
numbers.

If n is even the middle number in n observations is the
average of the (n/2)th and the (n/2+1)th observation
Measures of Central Tendency and Dispersion

The Median
Example 4
Find the median of each set of data.
a. 1, 2, 2, 3, 3, 4, 4, 5, 5
b. 1, 2, 2, 3, 3, 4, 4, 4, 5, 5
Answers
a. Me = 3 (the 5th number)
b. The average of 5th and 6th numbers:
3 4
Me
2

3.5
Measures of Central Tendency and Dispersion

The Median
Example 5
Find the median of the following:
3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7
Solution:
Arrange from lowest to highest:
2, 3, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10
The median is 7.
Measures of Central Tendency and Dispersion

The Mode
It is the number that appears most
frequently in a set of data.
 It is used when the most typical
(common) value is desired.
 It is not always unique. A distribution
can have no mode, one mode, or more
than one mode. When there are two or
more modes, we say the distribution is
multimodal.


(for two modes, we say that the distribution is
bimodal)
Measures of Central Tendency and Dispersion

The Mode
Example 6
The table shows the
scores of 20 students in
a 5-item AP quiz.

SCORE

NO. OF
STUDENTS

5

6

4

7

3

4

What is the modal
score?

2

2

Answer: 4

1

1
Measures of Central Tendency and Dispersion

The Mode
Example 7
Find the mode of each set of data.
a.

1, 2, 2, 3, 3, 4, 4, 4, 5, 5 Mo = 4

a.

1, 2, 2, 3, 3, 3,4, 4, 4, 5, 5Mo = 3 and 4

a.

1, 2, 3, 4, 5

No mode
Measures of Central Tendency and Dispersion

The Mode
Example 8
Find the mode of the following:
3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7
Solution:
Arrange from lowest to highest:
2, 3, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10
The mode is 7.
Measures of Central Tendency and Dispersion

Check your understanding
4, 8, 12, 15, 3, 2, 6, 9, 8, 7
The data set above gives the waiting times
(in minutes) of 10 students waiting for a
bus. Find the mean, median, and mode of
the data set.
Measures of Central Tendency and Dispersion

Check your understanding
4, 8, 12, 15, 3, 2, 6, 9, 8, 7
The data set above gives the waiting times (in minutes)
of 10 students waiting for a bus. Find the mean, median,
and mode of the data set.

Solution
Arrange the data first in increasing order:
2, 3, 4, 6, 7, 8, 8, 9, 12, 15
Mean : x

74
10

7.4 min

Median : Me

Mode : Mo 8 min

7 8
2

7.5 min
The Measures of
Dispersion
Measures of Central Tendency and Dispersion

Measures of Dispersion
The Measure of Dispersion or Variability
1. Range – the difference of the largest
and smallest value
2. Mean Absolute Deviation – the
average of the positive differences
from the mean
3. Standard deviation – involves the
average of the squared differences
from the mean.
(related: variance)
Measures of Central Tendency and Dispersion

Range
Simply the difference between the largest and
smallest values in a set of data
 Useful for analysis of fluctuations and for
ordinal data
 Is considered primitive as it considers only
the extreme values which may not be useful
indicators of the bulk of the population.
 The formula is:


Range = largest observation - smallest observation
Measures of Central Tendency and Dispersion

Range
Example 10
Find the range of the following data sets:
a.

3, 5, 10, 4, 3

range 10 3 7

b.

85, 87, 89, 90, 91, 98

range 98 85 13

c.

3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7
range 10 2 8
Measures of Central Tendency and Dispersion

Mean Deviation
It measures the ‗average‘ distance of each
observation away from the mean of the data
 Gives an equal weight to each observation
 Generally more sensitive than the range, since
a change in any value will affect it
 The formula is


MD

x x
n

x
where x is a quantity in the set,
and n is the number of data.

is the mean,
Measures of Central Tendency and Dispersion

Mean Deviation
To find the mean deviation:MD
1.

2.

x x
n

Compute the mean.
Get all the POSITIVE difference of each
number and the mean.
(It‘s the same as getting the absolute value of each difference)

3.
4.

Add all the results in step 2.
Divide by the number of data.
Measures of Central Tendency and Dispersion

Mean Deviation
Example 11
Find the mean deviation of
3, 6, 6, 7, 8, 11, 15, 16
Solution
x
STEP 1: Find the mean:

72
8

9
Measures of Central Tendency and Dispersion

Mean Deviation
VALUE

Example 11
Find the mean deviation of
3, 6, 6, 7, 8, 11, 15, 16

STEP 2: Find the
POSITIVE
difference of each
number and the
mean (9).

POSITIVE
DIFFERENCE

3
6

6
3

6
7
8
11
15
16

3
2
1
2
6
7
Measures of Central Tendency and Dispersion

Mean Deviation
VALUE

Example 11
Find the mean deviation of
3, 6, 6, 7, 8, 11, 15, 16

STEP 3: Add
all the
differences.

POSITIVE
DIFFERENCE

3
6

6
3

6
7
8
11
15
16

3
2
1
2
6
7

sum

30
Measures of Central Tendency and Dispersion

Mean Deviation
VALUE

Example 11
Find the mean deviation of
3, 6, 6, 7, 8, 11, 15, 16

STEP 4: Divide the result
by the number of data to
get the MD:

MD

30
8

3.75

POSITIVE
DIFFERENCE

3
6

6
3

6
7
8
11
15
16

3
2
1
2
6
7

sum

30
Measures of Central Tendency and Dispersion

Mean Deviation
What does the answer in the
previous example mean?
It means that the quantities have an average
difference of 3.75 from the mean (plus or
minus).
Measures of Central Tendency and Dispersion

Standard Deviation
Measures the variation of observations
from the mean
 The most common measure of
dispersion
 Takes into account every observation
 Measures the ‗average deviation‘ of
observations from the mean
 Works with squares of residuals, not
absolute values—easier to use in further

Measures of Central Tendency and Dispersion

Standard Deviation


The formula for the standard deviation
is
2
x x

n
where x is a quantity in the set,
x is the
mean, and n is the number of data.
Measures of Central Tendency and Dispersion

Variance
 The

variance is simply the square of
the standard deviation, or 2

Variance :

2

x x
n

2
Measures of Central Tendency and Dispersion

Standard Deviation
x x
To find the standard deviation:
n
1. Compute the mean.
2. Get the difference of each number and the
mean.
3. Square each difference
4. Add all the results in step 3.
5. Divide by the number of data.
6. Get the square root.
Note: If the VARIANCE is to be computed, skip
the last step.

2
Measures of Central Tendency and Dispersion

Standard Deviation
Population versus Sample Standard
Deviation
 The standard deviation used here is
called the POPULATION standard
deviation.
 For very large populations, the SAMPLE
standard deviation (s) is used. Its
2
formula is
x x

s

n 1
Measures of Central Tendency and Dispersion

Standard Deviation
Alternative Formula for the Standard
Deviation formula for standard deviation
 Another
uses only the sum of the data as well the
sum of the squares of the data. This is

n

x

2

x
n

2
Measures of Central Tendency and Dispersion

Standard Deviation
To find the standard deviation using the
alternative formula:
n x
x
n
1. Compute the squares of the data.
2. Get the sum of the data and the sum of the
squares of the data.
3. Multiply the sum of the squares by the
number of data, then subtract to the square
of the sum of the data.
4. Get the square root of the result in step 3.
5. Divide the result by the number of data.
2

2
Measures of Central Tendency and Dispersion

Standard Deviation
Example 12
Find the standard deviation of
3, 6, 6, 7, 8, 11, 15, 16
using the given and the alternative
formulas.
Solution

Before using the formulas, it‘s better to
tabulate all results.
Measures of Central Tendency and Dispersion

Standard Deviation
Using the given formula

x

x–x

x x

2

n

(x – x)2

3

–6

36

6
6
7
8
11
15
16

–3
–3
–2
–1
2
6
7
sum

9
9
4
1
4
36
49
148

x x
n

148
8

4.3

2
Measures of Central Tendency and Dispersion

Standard Deviation
Using the alternative formula

x
3

sum

x2
9

6
6
7
8
11
15
16
72

36
36
49
64
121
225
256
796

n

x2

x

2

n

n

x

2

x

2

n
8 796

72

2

8
1
,184
8

4.3

Ano ang
pipiliin
mo?
Measures of Central Tendency and Dispersion

Standard Deviation
Remark:
For both cases, the variance is simply
the square of the standard deviation.
The value2is 74
Woohoo…
Measures of Central Tendency and Dispersion

Check your understanding
Find the standard deviation
and variance of the following
data set:
4, 8, 12, 15, 3, 2, 6, 9, 8, 7
Measures of Central Tendency and Dispersion
Thank
you!

More Related Content

What's hot

4. parameter and statistic
4. parameter and statistic4. parameter and statistic
4. parameter and statistic
ONE Virtual Services
 
Geometric Mean
Geometric MeanGeometric Mean
Geometric Mean
Sumit Kumar
 
Lecture on Measures of Variability/Dispersion by RDSII
Lecture on Measures of Variability/Dispersion by RDSIILecture on Measures of Variability/Dispersion by RDSII
Lecture on Measures of Variability/Dispersion by RDSII
REYNALDO II SALAYOG
 
Chapter 3 260110 044503
Chapter 3 260110 044503Chapter 3 260110 044503
Chapter 3 260110 044503
guest25d353
 
6. point and interval estimation
6. point and interval estimation6. point and interval estimation
6. point and interval estimation
ONE Virtual Services
 
Elementary Statistics
Elementary Statistics Elementary Statistics
Elementary Statistics
jennytuazon01630
 
Advanced statistics
Advanced statisticsAdvanced statistics
Advanced statistics
Romel Villarubia
 
Measures of Variability
Measures of VariabilityMeasures of Variability
Measures of Variability
jasondroesch
 
Grouped Mean Median Mode
Grouped Mean Median ModeGrouped Mean Median Mode
Grouped Mean Median Mode
Passy World
 
MEASURES OF POSITION
MEASURES OF POSITIONMEASURES OF POSITION
MEASURES OF POSITION
Hazel Charise Gonzales
 
Measures of Variability
Measures of VariabilityMeasures of Variability
Measures of Variability
Mary Krystle Dawn Sulleza
 
The sampling distribution
The sampling distributionThe sampling distribution
The sampling distribution
Harve Abella
 
Quartile
QuartileQuartile
Quartile
Kemberly Lee
 
Measures of Variation
Measures of VariationMeasures of Variation
Measures of Variation
Rica Joy Pontilar
 
Inferential Statistics
Inferential StatisticsInferential Statistics
Inferential Statistics
ewhite00
 
Mean for Grouped Data
Mean for Grouped DataMean for Grouped Data
Measures of Central Tendency: Ungrouped and Grouped
Measures of Central Tendency: Ungrouped and GroupedMeasures of Central Tendency: Ungrouped and Grouped
Measures of Central Tendency: Ungrouped and Grouped
MaryGraceRecaaAgusti
 
What is a paired samples t test
What is a paired samples t testWhat is a paired samples t test
What is a paired samples t test
Ken Plummer
 
Measures of dispersion discuss 2.2
Measures of dispersion discuss 2.2Measures of dispersion discuss 2.2
Measures of dispersion discuss 2.2
Makati Science High School
 
Statistical treatment and data processing copy
Statistical treatment and data processing   copyStatistical treatment and data processing   copy
Statistical treatment and data processing copy
SWEET PEARL GAMAYON
 

What's hot (20)

4. parameter and statistic
4. parameter and statistic4. parameter and statistic
4. parameter and statistic
 
Geometric Mean
Geometric MeanGeometric Mean
Geometric Mean
 
Lecture on Measures of Variability/Dispersion by RDSII
Lecture on Measures of Variability/Dispersion by RDSIILecture on Measures of Variability/Dispersion by RDSII
Lecture on Measures of Variability/Dispersion by RDSII
 
Chapter 3 260110 044503
Chapter 3 260110 044503Chapter 3 260110 044503
Chapter 3 260110 044503
 
6. point and interval estimation
6. point and interval estimation6. point and interval estimation
6. point and interval estimation
 
Elementary Statistics
Elementary Statistics Elementary Statistics
Elementary Statistics
 
Advanced statistics
Advanced statisticsAdvanced statistics
Advanced statistics
 
Measures of Variability
Measures of VariabilityMeasures of Variability
Measures of Variability
 
Grouped Mean Median Mode
Grouped Mean Median ModeGrouped Mean Median Mode
Grouped Mean Median Mode
 
MEASURES OF POSITION
MEASURES OF POSITIONMEASURES OF POSITION
MEASURES OF POSITION
 
Measures of Variability
Measures of VariabilityMeasures of Variability
Measures of Variability
 
The sampling distribution
The sampling distributionThe sampling distribution
The sampling distribution
 
Quartile
QuartileQuartile
Quartile
 
Measures of Variation
Measures of VariationMeasures of Variation
Measures of Variation
 
Inferential Statistics
Inferential StatisticsInferential Statistics
Inferential Statistics
 
Mean for Grouped Data
Mean for Grouped DataMean for Grouped Data
Mean for Grouped Data
 
Measures of Central Tendency: Ungrouped and Grouped
Measures of Central Tendency: Ungrouped and GroupedMeasures of Central Tendency: Ungrouped and Grouped
Measures of Central Tendency: Ungrouped and Grouped
 
What is a paired samples t test
What is a paired samples t testWhat is a paired samples t test
What is a paired samples t test
 
Measures of dispersion discuss 2.2
Measures of dispersion discuss 2.2Measures of dispersion discuss 2.2
Measures of dispersion discuss 2.2
 
Statistical treatment and data processing copy
Statistical treatment and data processing   copyStatistical treatment and data processing   copy
Statistical treatment and data processing copy
 

Viewers also liked

Measures of central tendency
Measures of central tendencyMeasures of central tendency
Measures of central tendency
Chie Pegollo
 
CABT Math 8 - Angles and Angle Measurements
CABT Math 8 - Angles and Angle MeasurementsCABT Math 8 - Angles and Angle Measurements
CABT Math 8 - Angles and Angle Measurements
Gilbert Joseph Abueg
 
CABT Math 8 - Properties of Quadrilaterals
CABT Math 8 - Properties of QuadrilateralsCABT Math 8 - Properties of Quadrilaterals
CABT Math 8 - Properties of Quadrilaterals
Gilbert Joseph Abueg
 
Mean, Median, Mode: Measures of Central Tendency
Mean, Median, Mode: Measures of Central Tendency Mean, Median, Mode: Measures of Central Tendency
Mean, Median, Mode: Measures of Central Tendency
Jan Nah
 
K to 12 - Grade 8 Math Learner Module
K to 12 - Grade 8 Math Learner ModuleK to 12 - Grade 8 Math Learner Module
K to 12 - Grade 8 Math Learner Module
Nico Granada
 
CABT Math 8 - Basics of Geometry
CABT Math 8 - Basics of GeometryCABT Math 8 - Basics of Geometry
CABT Math 8 - Basics of Geometry
Gilbert Joseph Abueg
 
CABT Math 8 - Fundamental Principle of Counting
CABT Math 8 - Fundamental Principle of CountingCABT Math 8 - Fundamental Principle of Counting
CABT Math 8 - Fundamental Principle of Counting
Gilbert Joseph Abueg
 
Percentage Review Math 8
Percentage Review Math 8Percentage Review Math 8
Percentage Review Math 8
kbrach
 
Mathematics 8 Basic Concepts of Probability
Mathematics 8 Basic Concepts of ProbabilityMathematics 8 Basic Concepts of Probability
Mathematics 8 Basic Concepts of Probability
Juan Miguel Palero
 
Strategic Intervention Materials
Strategic Intervention MaterialsStrategic Intervention Materials
Strategic Intervention Materials
Brian Mary
 
K to 12 - Grade 8 Math Learners Module Quarter 2
K to 12 - Grade  8 Math Learners Module Quarter 2K to 12 - Grade  8 Math Learners Module Quarter 2
K to 12 - Grade 8 Math Learners Module Quarter 2
Nico Granada
 
K to 12 - Grade 8 Science Learner Module
K to 12 - Grade 8 Science Learner ModuleK to 12 - Grade 8 Science Learner Module
K to 12 - Grade 8 Science Learner Module
Nico Granada
 
10 math pa_l_08-04
10 math pa_l_08-0410 math pa_l_08-04
10 math pa_l_08-04
lkemper
 
Exponents, factors, and fractions math 7 chapter 2 review
Exponents, factors, and fractions  math 7 chapter 2 reviewExponents, factors, and fractions  math 7 chapter 2 review
Exponents, factors, and fractions math 7 chapter 2 review
lashwnb
 
Consumer Math Slides January 8, 2008
Consumer Math Slides January 8, 2008Consumer Math Slides January 8, 2008
Consumer Math Slides January 8, 2008
Darren Kuropatwa
 
Unit 8 Math Review
Unit 8 Math ReviewUnit 8 Math Review
Unit 8 Math Review
standrewmlewis
 
Second Quarter Group F Math Peta - Factoring (GCMF, DTS, STC, DTC, PST, QT1, ...
Second Quarter Group F Math Peta - Factoring (GCMF, DTS, STC, DTC, PST, QT1, ...Second Quarter Group F Math Peta - Factoring (GCMF, DTS, STC, DTC, PST, QT1, ...
Second Quarter Group F Math Peta - Factoring (GCMF, DTS, STC, DTC, PST, QT1, ...
GroupFMathPeta
 
Graphing, Slope & Intercepts
Graphing, Slope & InterceptsGraphing, Slope & Intercepts
Graphing, Slope & Intercepts
guest64649f
 
Lecture 19 section 8.1 system of equns
Lecture 19   section 8.1 system of equnsLecture 19   section 8.1 system of equns
Lecture 19 section 8.1 system of equns
njit-ronbrown
 
Measures of central tendecy
Measures of central tendecyMeasures of central tendecy
Measures of central tendecy
inahdiego
 

Viewers also liked (20)

Measures of central tendency
Measures of central tendencyMeasures of central tendency
Measures of central tendency
 
CABT Math 8 - Angles and Angle Measurements
CABT Math 8 - Angles and Angle MeasurementsCABT Math 8 - Angles and Angle Measurements
CABT Math 8 - Angles and Angle Measurements
 
CABT Math 8 - Properties of Quadrilaterals
CABT Math 8 - Properties of QuadrilateralsCABT Math 8 - Properties of Quadrilaterals
CABT Math 8 - Properties of Quadrilaterals
 
Mean, Median, Mode: Measures of Central Tendency
Mean, Median, Mode: Measures of Central Tendency Mean, Median, Mode: Measures of Central Tendency
Mean, Median, Mode: Measures of Central Tendency
 
K to 12 - Grade 8 Math Learner Module
K to 12 - Grade 8 Math Learner ModuleK to 12 - Grade 8 Math Learner Module
K to 12 - Grade 8 Math Learner Module
 
CABT Math 8 - Basics of Geometry
CABT Math 8 - Basics of GeometryCABT Math 8 - Basics of Geometry
CABT Math 8 - Basics of Geometry
 
CABT Math 8 - Fundamental Principle of Counting
CABT Math 8 - Fundamental Principle of CountingCABT Math 8 - Fundamental Principle of Counting
CABT Math 8 - Fundamental Principle of Counting
 
Percentage Review Math 8
Percentage Review Math 8Percentage Review Math 8
Percentage Review Math 8
 
Mathematics 8 Basic Concepts of Probability
Mathematics 8 Basic Concepts of ProbabilityMathematics 8 Basic Concepts of Probability
Mathematics 8 Basic Concepts of Probability
 
Strategic Intervention Materials
Strategic Intervention MaterialsStrategic Intervention Materials
Strategic Intervention Materials
 
K to 12 - Grade 8 Math Learners Module Quarter 2
K to 12 - Grade  8 Math Learners Module Quarter 2K to 12 - Grade  8 Math Learners Module Quarter 2
K to 12 - Grade 8 Math Learners Module Quarter 2
 
K to 12 - Grade 8 Science Learner Module
K to 12 - Grade 8 Science Learner ModuleK to 12 - Grade 8 Science Learner Module
K to 12 - Grade 8 Science Learner Module
 
10 math pa_l_08-04
10 math pa_l_08-0410 math pa_l_08-04
10 math pa_l_08-04
 
Exponents, factors, and fractions math 7 chapter 2 review
Exponents, factors, and fractions  math 7 chapter 2 reviewExponents, factors, and fractions  math 7 chapter 2 review
Exponents, factors, and fractions math 7 chapter 2 review
 
Consumer Math Slides January 8, 2008
Consumer Math Slides January 8, 2008Consumer Math Slides January 8, 2008
Consumer Math Slides January 8, 2008
 
Unit 8 Math Review
Unit 8 Math ReviewUnit 8 Math Review
Unit 8 Math Review
 
Second Quarter Group F Math Peta - Factoring (GCMF, DTS, STC, DTC, PST, QT1, ...
Second Quarter Group F Math Peta - Factoring (GCMF, DTS, STC, DTC, PST, QT1, ...Second Quarter Group F Math Peta - Factoring (GCMF, DTS, STC, DTC, PST, QT1, ...
Second Quarter Group F Math Peta - Factoring (GCMF, DTS, STC, DTC, PST, QT1, ...
 
Graphing, Slope & Intercepts
Graphing, Slope & InterceptsGraphing, Slope & Intercepts
Graphing, Slope & Intercepts
 
Lecture 19 section 8.1 system of equns
Lecture 19   section 8.1 system of equnsLecture 19   section 8.1 system of equns
Lecture 19 section 8.1 system of equns
 
Measures of central tendecy
Measures of central tendecyMeasures of central tendecy
Measures of central tendecy
 

Similar to CABT Math 8 measures of central tendency and dispersion

Descriptive Statistics: Mean, Median Mode and Standard Deviation.
Descriptive Statistics: Mean, Median Mode and Standard Deviation.Descriptive Statistics: Mean, Median Mode and Standard Deviation.
Descriptive Statistics: Mean, Median Mode and Standard Deviation.
Megha Sharma
 
Intro to Biostat. ppt
Intro to Biostat. pptIntro to Biostat. ppt
Intro to Biostat. ppt
AhmadYarSukhera
 
Basic Statistical Concepts in Machine Learning.pptx
Basic Statistical Concepts in Machine Learning.pptxBasic Statistical Concepts in Machine Learning.pptx
Basic Statistical Concepts in Machine Learning.pptx
bajajrishabh96tech
 
Stat11t chapter3
Stat11t chapter3Stat11t chapter3
Stat11t chapter3
raylenepotter
 
Data analysis
Data analysisData analysis
Data analysis
metalkid132
 
3. measures of central tendency
3. measures of central tendency3. measures of central tendency
3. measures of central tendency
renz50
 
Descriptions of data statistics for research
Descriptions of data   statistics for researchDescriptions of data   statistics for research
Descriptions of data statistics for research
Harve Abella
 
5.DATA SUMMERISATION.ppt
5.DATA SUMMERISATION.ppt5.DATA SUMMERISATION.ppt
5.DATA SUMMERISATION.ppt
chusematelephone
 
STATISTICAL PROCEDURES (Discriptive Statistics).pptx
STATISTICAL PROCEDURES (Discriptive Statistics).pptxSTATISTICAL PROCEDURES (Discriptive Statistics).pptx
STATISTICAL PROCEDURES (Discriptive Statistics).pptx
MuhammadNafees42
 
Descriptive Statistics.pptx
Descriptive Statistics.pptxDescriptive Statistics.pptx
Descriptive Statistics.pptx
Shashank Mishra
 
Measures of Central Tendency.pdf
Measures of Central Tendency.pdfMeasures of Central Tendency.pdf
Measures of Central Tendency.pdf
DenogieCortes
 
Bio statistics
Bio statisticsBio statistics
Bio statistics
Nc Das
 
Machine learning pre requisite
Machine learning pre requisiteMachine learning pre requisite
Machine learning pre requisite
Ram Singh
 
Statistics and permeability engineering reports
Statistics and permeability engineering reportsStatistics and permeability engineering reports
Statistics and permeability engineering reports
wwwmostafalaith99
 
Measures of central tendency and dispersion
Measures of central tendency and dispersionMeasures of central tendency and dispersion
Measures of central tendency and dispersion
Abhinav yadav
 
Chapter 11 Psrm
Chapter 11 PsrmChapter 11 Psrm
Chapter 11 Psrm
mandrewmartin
 
LESSON-8-ANALYSIS-INTERPRETATION-AND-USE-OF-TEST-DATA.pptx
LESSON-8-ANALYSIS-INTERPRETATION-AND-USE-OF-TEST-DATA.pptxLESSON-8-ANALYSIS-INTERPRETATION-AND-USE-OF-TEST-DATA.pptx
LESSON-8-ANALYSIS-INTERPRETATION-AND-USE-OF-TEST-DATA.pptx
MarjoriAnneDelosReye
 
3 descritive statistics measure of central tendency variatio
3 descritive statistics measure of   central   tendency variatio3 descritive statistics measure of   central   tendency variatio
3 descritive statistics measure of central tendency variatio
Lama K Banna
 
Measures of central tendency
Measures of central tendencyMeasures of central tendency
Measures of central tendency
Mmedsc Hahm
 
UNIT I -Data and Data Collection1.ppt
UNIT I -Data and Data Collection1.pptUNIT I -Data and Data Collection1.ppt
UNIT I -Data and Data Collection1.ppt
NAGESH108233
 

Similar to CABT Math 8 measures of central tendency and dispersion (20)

Descriptive Statistics: Mean, Median Mode and Standard Deviation.
Descriptive Statistics: Mean, Median Mode and Standard Deviation.Descriptive Statistics: Mean, Median Mode and Standard Deviation.
Descriptive Statistics: Mean, Median Mode and Standard Deviation.
 
Intro to Biostat. ppt
Intro to Biostat. pptIntro to Biostat. ppt
Intro to Biostat. ppt
 
Basic Statistical Concepts in Machine Learning.pptx
Basic Statistical Concepts in Machine Learning.pptxBasic Statistical Concepts in Machine Learning.pptx
Basic Statistical Concepts in Machine Learning.pptx
 
Stat11t chapter3
Stat11t chapter3Stat11t chapter3
Stat11t chapter3
 
Data analysis
Data analysisData analysis
Data analysis
 
3. measures of central tendency
3. measures of central tendency3. measures of central tendency
3. measures of central tendency
 
Descriptions of data statistics for research
Descriptions of data   statistics for researchDescriptions of data   statistics for research
Descriptions of data statistics for research
 
5.DATA SUMMERISATION.ppt
5.DATA SUMMERISATION.ppt5.DATA SUMMERISATION.ppt
5.DATA SUMMERISATION.ppt
 
STATISTICAL PROCEDURES (Discriptive Statistics).pptx
STATISTICAL PROCEDURES (Discriptive Statistics).pptxSTATISTICAL PROCEDURES (Discriptive Statistics).pptx
STATISTICAL PROCEDURES (Discriptive Statistics).pptx
 
Descriptive Statistics.pptx
Descriptive Statistics.pptxDescriptive Statistics.pptx
Descriptive Statistics.pptx
 
Measures of Central Tendency.pdf
Measures of Central Tendency.pdfMeasures of Central Tendency.pdf
Measures of Central Tendency.pdf
 
Bio statistics
Bio statisticsBio statistics
Bio statistics
 
Machine learning pre requisite
Machine learning pre requisiteMachine learning pre requisite
Machine learning pre requisite
 
Statistics and permeability engineering reports
Statistics and permeability engineering reportsStatistics and permeability engineering reports
Statistics and permeability engineering reports
 
Measures of central tendency and dispersion
Measures of central tendency and dispersionMeasures of central tendency and dispersion
Measures of central tendency and dispersion
 
Chapter 11 Psrm
Chapter 11 PsrmChapter 11 Psrm
Chapter 11 Psrm
 
LESSON-8-ANALYSIS-INTERPRETATION-AND-USE-OF-TEST-DATA.pptx
LESSON-8-ANALYSIS-INTERPRETATION-AND-USE-OF-TEST-DATA.pptxLESSON-8-ANALYSIS-INTERPRETATION-AND-USE-OF-TEST-DATA.pptx
LESSON-8-ANALYSIS-INTERPRETATION-AND-USE-OF-TEST-DATA.pptx
 
3 descritive statistics measure of central tendency variatio
3 descritive statistics measure of   central   tendency variatio3 descritive statistics measure of   central   tendency variatio
3 descritive statistics measure of central tendency variatio
 
Measures of central tendency
Measures of central tendencyMeasures of central tendency
Measures of central tendency
 
UNIT I -Data and Data Collection1.ppt
UNIT I -Data and Data Collection1.pptUNIT I -Data and Data Collection1.ppt
UNIT I -Data and Data Collection1.ppt
 

More from Gilbert Joseph Abueg

Math Reviewer - Word Problems in Algebra
Math Reviewer - Word Problems in AlgebraMath Reviewer - Word Problems in Algebra
Math Reviewer - Word Problems in Algebra
Gilbert Joseph Abueg
 
Math-tanong CEER 2012 - Set 2
Math-tanong CEER 2012 - Set 2Math-tanong CEER 2012 - Set 2
Math-tanong CEER 2012 - Set 2
Gilbert Joseph Abueg
 
Math-tanong CEER 2012 - Set 1 Solutions
Math-tanong CEER 2012 - Set 1 SolutionsMath-tanong CEER 2012 - Set 1 Solutions
Math-tanong CEER 2012 - Set 1 Solutions
Gilbert Joseph Abueg
 
Math-tanong CEER 2012 - Set 1 solutions
Math-tanong CEER 2012 - Set 1 solutionsMath-tanong CEER 2012 - Set 1 solutions
Math-tanong CEER 2012 - Set 1 solutions
Gilbert Joseph Abueg
 
Math-tanong CEER 2012 - Set 1
Math-tanong CEER 2012 - Set 1Math-tanong CEER 2012 - Set 1
Math-tanong CEER 2012 - Set 1
Gilbert Joseph Abueg
 
CEER 2012 Math Lecture
CEER 2012 Math LectureCEER 2012 Math Lecture
CEER 2012 Math Lecture
Gilbert Joseph Abueg
 

More from Gilbert Joseph Abueg (6)

Math Reviewer - Word Problems in Algebra
Math Reviewer - Word Problems in AlgebraMath Reviewer - Word Problems in Algebra
Math Reviewer - Word Problems in Algebra
 
Math-tanong CEER 2012 - Set 2
Math-tanong CEER 2012 - Set 2Math-tanong CEER 2012 - Set 2
Math-tanong CEER 2012 - Set 2
 
Math-tanong CEER 2012 - Set 1 Solutions
Math-tanong CEER 2012 - Set 1 SolutionsMath-tanong CEER 2012 - Set 1 Solutions
Math-tanong CEER 2012 - Set 1 Solutions
 
Math-tanong CEER 2012 - Set 1 solutions
Math-tanong CEER 2012 - Set 1 solutionsMath-tanong CEER 2012 - Set 1 solutions
Math-tanong CEER 2012 - Set 1 solutions
 
Math-tanong CEER 2012 - Set 1
Math-tanong CEER 2012 - Set 1Math-tanong CEER 2012 - Set 1
Math-tanong CEER 2012 - Set 1
 
CEER 2012 Math Lecture
CEER 2012 Math LectureCEER 2012 Math Lecture
CEER 2012 Math Lecture
 

Recently uploaded

Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumPhilippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum
MJDuyan
 
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
Nguyen Thanh Tu Collection
 
BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...
BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...
BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...
Nguyen Thanh Tu Collection
 
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPLAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
RAHUL
 
Advanced Java[Extra Concepts, Not Difficult].docx
Advanced Java[Extra Concepts, Not Difficult].docxAdvanced Java[Extra Concepts, Not Difficult].docx
Advanced Java[Extra Concepts, Not Difficult].docx
adhitya5119
 
Hindi varnamala | hindi alphabet PPT.pdf
Hindi varnamala | hindi alphabet PPT.pdfHindi varnamala | hindi alphabet PPT.pdf
Hindi varnamala | hindi alphabet PPT.pdf
Dr. Mulla Adam Ali
 
Wound healing PPT
Wound healing PPTWound healing PPT
Wound healing PPT
Jyoti Chand
 
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdfবাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
eBook.com.bd (প্রয়োজনীয় বাংলা বই)
 
Walmart Business+ and Spark Good for Nonprofits.pdf
Walmart Business+ and Spark Good for Nonprofits.pdfWalmart Business+ and Spark Good for Nonprofits.pdf
Walmart Business+ and Spark Good for Nonprofits.pdf
TechSoup
 
Main Java[All of the Base Concepts}.docx
Main Java[All of the Base Concepts}.docxMain Java[All of the Base Concepts}.docx
Main Java[All of the Base Concepts}.docx
adhitya5119
 
clinical examination of hip joint (1).pdf
clinical examination of hip joint (1).pdfclinical examination of hip joint (1).pdf
clinical examination of hip joint (1).pdf
Priyankaranawat4
 
spot a liar (Haiqa 146).pptx Technical writhing and presentation skills
spot a liar (Haiqa 146).pptx Technical writhing and presentation skillsspot a liar (Haiqa 146).pptx Technical writhing and presentation skills
spot a liar (Haiqa 146).pptx Technical writhing and presentation skills
haiqairshad
 
Présentationvvvvvvvvvvvvvvvvvvvvvvvvvvvv2.pptx
Présentationvvvvvvvvvvvvvvvvvvvvvvvvvvvv2.pptxPrésentationvvvvvvvvvvvvvvvvvvvvvvvvvvvv2.pptx
Présentationvvvvvvvvvvvvvvvvvvvvvvvvvvvv2.pptx
siemaillard
 
Leveraging Generative AI to Drive Nonprofit Innovation
Leveraging Generative AI to Drive Nonprofit InnovationLeveraging Generative AI to Drive Nonprofit Innovation
Leveraging Generative AI to Drive Nonprofit Innovation
TechSoup
 
UGC NET Exam Paper 1- Unit 1:Teaching Aptitude
UGC NET Exam Paper 1- Unit 1:Teaching AptitudeUGC NET Exam Paper 1- Unit 1:Teaching Aptitude
UGC NET Exam Paper 1- Unit 1:Teaching Aptitude
S. Raj Kumar
 
math operations ued in python and all used
math operations ued in python and all usedmath operations ued in python and all used
math operations ued in python and all used
ssuser13ffe4
 
ZK on Polkadot zero knowledge proofs - sub0.pptx
ZK on Polkadot zero knowledge proofs - sub0.pptxZK on Polkadot zero knowledge proofs - sub0.pptx
ZK on Polkadot zero knowledge proofs - sub0.pptx
dot55audits
 
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
PECB
 
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem studentsRHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
Himanshu Rai
 
B. Ed Syllabus for babasaheb ambedkar education university.pdf
B. Ed Syllabus for babasaheb ambedkar education university.pdfB. Ed Syllabus for babasaheb ambedkar education university.pdf
B. Ed Syllabus for babasaheb ambedkar education university.pdf
BoudhayanBhattachari
 

Recently uploaded (20)

Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumPhilippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum
 
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
 
BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...
BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...
BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...
 
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPLAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
 
Advanced Java[Extra Concepts, Not Difficult].docx
Advanced Java[Extra Concepts, Not Difficult].docxAdvanced Java[Extra Concepts, Not Difficult].docx
Advanced Java[Extra Concepts, Not Difficult].docx
 
Hindi varnamala | hindi alphabet PPT.pdf
Hindi varnamala | hindi alphabet PPT.pdfHindi varnamala | hindi alphabet PPT.pdf
Hindi varnamala | hindi alphabet PPT.pdf
 
Wound healing PPT
Wound healing PPTWound healing PPT
Wound healing PPT
 
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdfবাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
 
Walmart Business+ and Spark Good for Nonprofits.pdf
Walmart Business+ and Spark Good for Nonprofits.pdfWalmart Business+ and Spark Good for Nonprofits.pdf
Walmart Business+ and Spark Good for Nonprofits.pdf
 
Main Java[All of the Base Concepts}.docx
Main Java[All of the Base Concepts}.docxMain Java[All of the Base Concepts}.docx
Main Java[All of the Base Concepts}.docx
 
clinical examination of hip joint (1).pdf
clinical examination of hip joint (1).pdfclinical examination of hip joint (1).pdf
clinical examination of hip joint (1).pdf
 
spot a liar (Haiqa 146).pptx Technical writhing and presentation skills
spot a liar (Haiqa 146).pptx Technical writhing and presentation skillsspot a liar (Haiqa 146).pptx Technical writhing and presentation skills
spot a liar (Haiqa 146).pptx Technical writhing and presentation skills
 
Présentationvvvvvvvvvvvvvvvvvvvvvvvvvvvv2.pptx
Présentationvvvvvvvvvvvvvvvvvvvvvvvvvvvv2.pptxPrésentationvvvvvvvvvvvvvvvvvvvvvvvvvvvv2.pptx
Présentationvvvvvvvvvvvvvvvvvvvvvvvvvvvv2.pptx
 
Leveraging Generative AI to Drive Nonprofit Innovation
Leveraging Generative AI to Drive Nonprofit InnovationLeveraging Generative AI to Drive Nonprofit Innovation
Leveraging Generative AI to Drive Nonprofit Innovation
 
UGC NET Exam Paper 1- Unit 1:Teaching Aptitude
UGC NET Exam Paper 1- Unit 1:Teaching AptitudeUGC NET Exam Paper 1- Unit 1:Teaching Aptitude
UGC NET Exam Paper 1- Unit 1:Teaching Aptitude
 
math operations ued in python and all used
math operations ued in python and all usedmath operations ued in python and all used
math operations ued in python and all used
 
ZK on Polkadot zero knowledge proofs - sub0.pptx
ZK on Polkadot zero knowledge proofs - sub0.pptxZK on Polkadot zero knowledge proofs - sub0.pptx
ZK on Polkadot zero knowledge proofs - sub0.pptx
 
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
 
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem studentsRHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
RHEOLOGY Physical pharmaceutics-II notes for B.pharm 4th sem students
 
B. Ed Syllabus for babasaheb ambedkar education university.pdf
B. Ed Syllabus for babasaheb ambedkar education university.pdfB. Ed Syllabus for babasaheb ambedkar education university.pdf
B. Ed Syllabus for babasaheb ambedkar education university.pdf
 

CABT Math 8 measures of central tendency and dispersion

  • 1.
  • 2.
  • 3.
  • 4. A Brief Introduction to Statistics What is Statistics What is Statistics? 1. The science that deals with the collection, organization, presentation, analysis, and interpretation of numerical data to obtain useful and meaningful information 2. A collection of quantitative data pertaining to a subject or group. Examples are blood pressure statistics etc.
  • 5. A Brief Introduction to Statistics Branches of Statistics Two branches of statistics: 1. Descriptive Statistics: Describes the characteristics of a product or process using information collected on it. 2. Inferential Statistics (Inductive): Draws conclusions on unknown process parameters based on information contained in a sample. Uses probability
  • 6. A Brief Introduction to Statistics Data DATA is any quantitative or qualitative information. Types of Data: 1. Quantitative – numerical information obtained from counting or measuring (e.g. age, qtr. exam scores, height) 2. Qualitative – descriptive attributes that cannot be subjected to mathematical operations (e.g. gender, religion, citizenship)
  • 7. Measures of Central Tendency and Dispersion The Measures of Central Tendency and Dispersion Statistics use numerical values used to summarize and compare sets of data.  Measure of Central Tendency: number used to represent the center or middle set of a set of data  Measure of Dispersion (or Variability): refers to the spread of values about the mean.  (i.e., how spread out the values are with respect to the mean)
  • 9. Measures of Central Tendency and Dispersion Measures of Central Tendency The Measure of Central Tendency: 1. Mean - the (arithmetic) average (or the sum of the quantities divided by the number of quantities) Median – the middle value of a set of ordered data 3. Mode – number in a data set that occurs most frequently 2.
  • 10. Measures of Central Tendency and Dispersion The Mean It‘s known as the typical ―average.‖  It is the most common measure of central tendency.  Symbolized as: ◦ x for the mean of a sample ◦ μ (Greek letter mu) for the mean of a population • It‘s equal to the sum of the quantities in the data set divided by the number of quantities  x x n
  • 11. Measures of Central Tendency and Dispersion The Mean Example 1 Find the mean of the numbers in the following data sets: a. b. 3, 5, 10, 4, 3 x 3 5 10 4 3 5 85, 87, 89, 90, 91, 98 x 540 6 90 25 5 5
  • 12. Measures of Central Tendency and Dispersion The Mean Example 2 The table on the right shows the age of 13 applicants for a job in a factory in EPZA. What is the average age of the applicants? (Adapted from DOLE-BLES i-Learnstat module on Measures of Central Tendency) Solution: x 318 13 24.5
  • 13. Measures of Central Tendency and Dispersion The Weighted Mean It is a mean where some values contribute more than others.  Each quantity is assigned a corresponding WEIGHT  (e.g. frequency or number, units, per cent)  The weighted mean is equal to the sum of the products of the quantities (x) and their corresponding weights (w), divided by the sum of the weights. x wx w
  • 14. Measures of Central Tendency and Dispersion The Weighted Mean Example 3 SCORE NO. OF STUDENTS 5 8 4 6 3 3 2 2 1 1 The table shows the scores of 20 students in a 5-item Math IV seatwork. Find the average score of the class.
  • 15. Measures of Central Tendency and Dispersion The Weighted Mean Example 3 SolutionSCORE x 78 20 3.9 PRODUCT 5 8 40 4 6 24 3 3 9 2 2 4 1 Multiply the scores by the number of students, then find the sum. Finally, divide by the total number of students The average score is NO. OF STUDENTS 1 1 sums 20 78
  • 16. Measures of Central Tendency and Dispersion The Median Used to find the middle value (center) of a distribution.  Used when one must determine whether the data values fall into either the upper 50% or lower 50% of a distribution.  Used when one needs to report the typical value of a data set, ignoring the outliers (few extreme values in a data set).  ◦ Example: median salary, median home prices in a market
  • 17. Measures of Central Tendency and Dispersion The Median How to find the median:   Order the data in increasing order. If the number of data is ODD, the median is the middle number. If n is odd, the middle number in n observations is the (n + 1)/2 th observation  If the number of data is EVEN, the median is the mean of the two middle numbers. If n is even the middle number in n observations is the average of the (n/2)th and the (n/2+1)th observation
  • 18. Measures of Central Tendency and Dispersion The Median Example 4 Find the median of each set of data. a. 1, 2, 2, 3, 3, 4, 4, 5, 5 b. 1, 2, 2, 3, 3, 4, 4, 4, 5, 5 Answers a. Me = 3 (the 5th number) b. The average of 5th and 6th numbers: 3 4 Me 2 3.5
  • 19. Measures of Central Tendency and Dispersion The Median Example 5 Find the median of the following: 3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7 Solution: Arrange from lowest to highest: 2, 3, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10 The median is 7.
  • 20. Measures of Central Tendency and Dispersion The Mode It is the number that appears most frequently in a set of data.  It is used when the most typical (common) value is desired.  It is not always unique. A distribution can have no mode, one mode, or more than one mode. When there are two or more modes, we say the distribution is multimodal.  (for two modes, we say that the distribution is bimodal)
  • 21. Measures of Central Tendency and Dispersion The Mode Example 6 The table shows the scores of 20 students in a 5-item AP quiz. SCORE NO. OF STUDENTS 5 6 4 7 3 4 What is the modal score? 2 2 Answer: 4 1 1
  • 22. Measures of Central Tendency and Dispersion The Mode Example 7 Find the mode of each set of data. a. 1, 2, 2, 3, 3, 4, 4, 4, 5, 5 Mo = 4 a. 1, 2, 2, 3, 3, 3,4, 4, 4, 5, 5Mo = 3 and 4 a. 1, 2, 3, 4, 5 No mode
  • 23. Measures of Central Tendency and Dispersion The Mode Example 8 Find the mode of the following: 3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7 Solution: Arrange from lowest to highest: 2, 3, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10 The mode is 7.
  • 24. Measures of Central Tendency and Dispersion Check your understanding 4, 8, 12, 15, 3, 2, 6, 9, 8, 7 The data set above gives the waiting times (in minutes) of 10 students waiting for a bus. Find the mean, median, and mode of the data set.
  • 25. Measures of Central Tendency and Dispersion Check your understanding 4, 8, 12, 15, 3, 2, 6, 9, 8, 7 The data set above gives the waiting times (in minutes) of 10 students waiting for a bus. Find the mean, median, and mode of the data set. Solution Arrange the data first in increasing order: 2, 3, 4, 6, 7, 8, 8, 9, 12, 15 Mean : x 74 10 7.4 min Median : Me Mode : Mo 8 min 7 8 2 7.5 min
  • 27. Measures of Central Tendency and Dispersion Measures of Dispersion The Measure of Dispersion or Variability 1. Range – the difference of the largest and smallest value 2. Mean Absolute Deviation – the average of the positive differences from the mean 3. Standard deviation – involves the average of the squared differences from the mean. (related: variance)
  • 28. Measures of Central Tendency and Dispersion Range Simply the difference between the largest and smallest values in a set of data  Useful for analysis of fluctuations and for ordinal data  Is considered primitive as it considers only the extreme values which may not be useful indicators of the bulk of the population.  The formula is:  Range = largest observation - smallest observation
  • 29. Measures of Central Tendency and Dispersion Range Example 10 Find the range of the following data sets: a. 3, 5, 10, 4, 3 range 10 3 7 b. 85, 87, 89, 90, 91, 98 range 98 85 13 c. 3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7 range 10 2 8
  • 30. Measures of Central Tendency and Dispersion Mean Deviation It measures the ‗average‘ distance of each observation away from the mean of the data  Gives an equal weight to each observation  Generally more sensitive than the range, since a change in any value will affect it  The formula is  MD x x n x where x is a quantity in the set, and n is the number of data. is the mean,
  • 31. Measures of Central Tendency and Dispersion Mean Deviation To find the mean deviation:MD 1. 2. x x n Compute the mean. Get all the POSITIVE difference of each number and the mean. (It‘s the same as getting the absolute value of each difference) 3. 4. Add all the results in step 2. Divide by the number of data.
  • 32. Measures of Central Tendency and Dispersion Mean Deviation Example 11 Find the mean deviation of 3, 6, 6, 7, 8, 11, 15, 16 Solution x STEP 1: Find the mean: 72 8 9
  • 33. Measures of Central Tendency and Dispersion Mean Deviation VALUE Example 11 Find the mean deviation of 3, 6, 6, 7, 8, 11, 15, 16 STEP 2: Find the POSITIVE difference of each number and the mean (9). POSITIVE DIFFERENCE 3 6 6 3 6 7 8 11 15 16 3 2 1 2 6 7
  • 34. Measures of Central Tendency and Dispersion Mean Deviation VALUE Example 11 Find the mean deviation of 3, 6, 6, 7, 8, 11, 15, 16 STEP 3: Add all the differences. POSITIVE DIFFERENCE 3 6 6 3 6 7 8 11 15 16 3 2 1 2 6 7 sum 30
  • 35. Measures of Central Tendency and Dispersion Mean Deviation VALUE Example 11 Find the mean deviation of 3, 6, 6, 7, 8, 11, 15, 16 STEP 4: Divide the result by the number of data to get the MD: MD 30 8 3.75 POSITIVE DIFFERENCE 3 6 6 3 6 7 8 11 15 16 3 2 1 2 6 7 sum 30
  • 36. Measures of Central Tendency and Dispersion Mean Deviation What does the answer in the previous example mean? It means that the quantities have an average difference of 3.75 from the mean (plus or minus).
  • 37. Measures of Central Tendency and Dispersion Standard Deviation Measures the variation of observations from the mean  The most common measure of dispersion  Takes into account every observation  Measures the ‗average deviation‘ of observations from the mean  Works with squares of residuals, not absolute values—easier to use in further 
  • 38. Measures of Central Tendency and Dispersion Standard Deviation  The formula for the standard deviation is 2 x x n where x is a quantity in the set, x is the mean, and n is the number of data.
  • 39. Measures of Central Tendency and Dispersion Variance  The variance is simply the square of the standard deviation, or 2 Variance : 2 x x n 2
  • 40. Measures of Central Tendency and Dispersion Standard Deviation x x To find the standard deviation: n 1. Compute the mean. 2. Get the difference of each number and the mean. 3. Square each difference 4. Add all the results in step 3. 5. Divide by the number of data. 6. Get the square root. Note: If the VARIANCE is to be computed, skip the last step. 2
  • 41. Measures of Central Tendency and Dispersion Standard Deviation Population versus Sample Standard Deviation  The standard deviation used here is called the POPULATION standard deviation.  For very large populations, the SAMPLE standard deviation (s) is used. Its 2 formula is x x s n 1
  • 42. Measures of Central Tendency and Dispersion Standard Deviation Alternative Formula for the Standard Deviation formula for standard deviation  Another uses only the sum of the data as well the sum of the squares of the data. This is n x 2 x n 2
  • 43. Measures of Central Tendency and Dispersion Standard Deviation To find the standard deviation using the alternative formula: n x x n 1. Compute the squares of the data. 2. Get the sum of the data and the sum of the squares of the data. 3. Multiply the sum of the squares by the number of data, then subtract to the square of the sum of the data. 4. Get the square root of the result in step 3. 5. Divide the result by the number of data. 2 2
  • 44. Measures of Central Tendency and Dispersion Standard Deviation Example 12 Find the standard deviation of 3, 6, 6, 7, 8, 11, 15, 16 using the given and the alternative formulas. Solution Before using the formulas, it‘s better to tabulate all results.
  • 45. Measures of Central Tendency and Dispersion Standard Deviation Using the given formula x x–x x x 2 n (x – x)2 3 –6 36 6 6 7 8 11 15 16 –3 –3 –2 –1 2 6 7 sum 9 9 4 1 4 36 49 148 x x n 148 8 4.3 2
  • 46. Measures of Central Tendency and Dispersion Standard Deviation Using the alternative formula x 3 sum x2 9 6 6 7 8 11 15 16 72 36 36 49 64 121 225 256 796 n x2 x 2 n n x 2 x 2 n 8 796 72 2 8 1 ,184 8 4.3 Ano ang pipiliin mo?
  • 47. Measures of Central Tendency and Dispersion Standard Deviation Remark: For both cases, the variance is simply the square of the standard deviation. The value2is 74 Woohoo…
  • 48. Measures of Central Tendency and Dispersion Check your understanding Find the standard deviation and variance of the following data set: 4, 8, 12, 15, 3, 2, 6, 9, 8, 7
  • 49. Measures of Central Tendency and Dispersion