Statistics & Probability

1. Measures of Central Tendency
Mean,Median and Mode
for Ungrouped Data
Basic Statistics

2. Measures of Central Tendency
In layman’s term, a measure
of central tendency is an AVERAGE.
It is a single number of value which
can be considered typical in a set of
data as a whole.
For example, in a class of 40
students, the average height would
be the typical height of the
members of this class as a whole.
Measures of central tendency are numerical descriptive
measures which indicate or locate the center
of a distribution or data set.

3. MEAN
Among the three measures of central tendency, the
mean is the most popular and widely used. It is sometimes
called the arithmetic mean.
If we compute the mean of the population, we call it
the parametric or population mean, denoted by μ
(read “mu”).
If we get the mean of the sample, we call it the
sample mean and it is denoted by (read “x bar”).
The MEAN of a set of values or measurements is the
sum of all the measurements divided by the number of
measurements in the set.

4. Mean for Ungrouped Data
Example 1:
Ms. Sulit collects the data on the ages of Mathematics teachers in
Santa Rosa School, and her study yields the following:
38 35 28 36 35 33 40
Solution:
= 35
Based on the computed mean, 38 is the average age of
Mathematics teachers in SRS.
For ungrouped or raw data, the mean has the following
formula.
where = mean
= sum of the measurements or values
n = number of measurements

5. Your turn!
Mang John is a meat vendor. The following are his sales for
the past six days. Compute his daily mean sales.
Tuesday P 5 800
Wednesday 8 600
Thursday 6 500
Friday 4 300
Saturday 12 500
Sunday 13 400
Solution:
= 51, 100
The average daily sales of Mang John is P51,100.

6. Weighted Mean
Weighted mean is the mean of a set of values wherein
each value or measurement has a different weight or
degree of importance. The following is its formula:
where = mean
x = measurement or value
w = number of measurements

7. Example
Below are Amaya’s subjects and the corresponding number
of units and grades she got for the previous grading
period. Compute her grade point average.
Subject Units Grade
Filipino .9 86
English 1.5 85
Mathematics 1.5 88
Science 1.8 87
Social Studies .9 86
TLE 1.2 83
MAPEH 1.2 87
= 86.1
Amaya’s average grade is 86.1

8. Your turn!
James obtained the following grades in his five subjects for
the second grading period. Compute his grade point average.
Solution:
= 87.67
James general average is 87.67
Subject Units Grade
Math 1.5 90
English 1.5 86
Science 1.8 88
Filipino 0.9 87
MAKABAYAN 1.5 87

9. Likert-type Question
Next are examples of Likert-type statements. Respondents
will choose the number which best represents their
feeling regarding the statements. Note that the
statements are grouped according to a theme.
This is used if the researcher wants to know the
feelings or opinions of the respondents regarding any topic or
issues of interest.
Choices
5 (SA) Strongly Agree
4 (A) Agree
3 (N) Neutral
2 (D) Disagree
1 (SD) Strongly Disagree

10. Students’ personal confidence in learning
Statistics
5 4 3 2 1
1. I am sure that I can learn Statistics
2. I think I can handle difficult lessons in
Statistics.
3. I can get good grades in Statistics.
Source: B.E. Blay, Elementary Statistics
Below are the responses in the Likert-type of
statements above. The table below shows the mean
responses and their interpretation. Using the formula for
computing the weighted mean, check the correctness of the
given means on the table.
5 4 3 2 1 Mean Interpretation
1 36 51 18 0 1 4.14 Agree
2 18 44 37 8 1 3.65 Agree
3 18 48 28 0 1 3.86 Agree

11. Likert-type Mean Interpretation
1.0 - 1.79 - Strongly Disagree
1.8 - 2.59 - Disagree
2.6 - 3.39 - Neutral
3.4 - 4.19 - Agree
4.2 - 5.00 - Strongly Agree

12. Your turn!
Below is the result of the responses to the following Likert-
type statements . Solve for the mean and give the
interpretation.
5 4 3 2 1 Mean Interpretation
1 33 49 26 1 1
2 35 45 31 0 1
3 34 58 21 0 0
Students’ perception on Statistics as a
subject
5 4 3 2 1
1. I think Statistics is a worthwhile, necessary
subject
2. I will use Statistics in many ways as a
professional
3. I’ll need a good understanding of Statistics
for my research work

13. Properties of Mean
1. Mean can be calculated for any set of
numerical data, so it always exists.
2. A set of numerical data has one and only one
mean.
3. Mean is the most reliable measure of central
tendency since it takes into account every item
in the set of data.
4. It is greatly affected by extreme or deviant
values (outliers)
5. It is used only if the data are interval or ratio.

14. MEDIAN
The MEDIAN, denoted Md, is the middle value of the
sample when the data are ranked in order according to size.
16 17 18 19 20 21 22
Median
16 17 18 19 20 21 22 23
Median

15. Your turn!
Compute the median and interpret the result.
1. In a survey of small businesses in Tondo, 10 bakeries
report the following numbers of employees:
15, 14, 12, 19, 13, 14 15, 18, 13, 19.
2. The random savings of 2nd year high school students
reveal the following current balances in their bank
accounts:
3. The following are the lifetimes of 9 lightbulbs in
thousands of hours.
Students A B C D E F G H
Current Balances P340 350 450 500 360 760 800 740
Lightbulb A B C D E F G H I
Lifetime 1.1 1.1 1.2 1.1 1.4 .9 .2 1.2 1.7

16. Properties of Median
1. Median is the score or class in the distribution
wherein 50% of the score fall below it and
another 50% lie.
2. Median is not affected by extreme or deviant
values.
3. Median is appropriate to use when there are
extreme or deviant values.
4. Median is used when the data are ordinal.
5. Median exists in both quantitative or qualitative
data.

17. MODE
The MODE, denoted Mo, is the value which occurs
most frequently in a set of measurements or values. In other
words, it is the most popular value in a given set.
Examples:
Find the Mode.
1. The ages of five students are: 17, 18, 23, 20, and 19
2. The following are the descriptive evaluations of 5
teachers: VS, S, VS, VS, O
3. The grades of five students are : 4.0, 3.5, 4.0, 3.5, and
1.0
4. The weights of five boys in pounds are: 117, 218, 233,
120, and 117

18. Properties
1. It is used when you want to find the value
which occurs most often.
2. It is a quick approximation of the average.
3. It is an inspection average.
4. It is the most unreliable among the three
measures of central tendency because its
value is undefined in some observations.

19. Your turn!
Find the mode and interpret it.
1. The following table shows the frequency of errors
committed by 10 typists per minute.
2. A random sample of 8 mango trees reveals the
following number of fruits they yield
3. The following are the scores of 9 students in a
Mathematics quiz.: 12, 15, 12, 8, 7, 15, 19, 24, 13
Mango Tree A B C D E F G H
No. of fruits 80 70 80 90 82 82 90 82
Typists A B C D E F G H I J
No. of errors per min. 5 3 3 7 2 8 8 4 7 10