Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.3: Measures of Relative Standing and Boxplots
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 2: Exploring Data with Tables and Graphs
2.1: Frequency Distributions for Organizing and Summarizing Data
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 2: Exploring Data with Tables and Graphs
2.2: Histograms
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 2: Exploring Data with Tables and Graphs
2.3: Graphs that Enlighten and Graphs that Deceive
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.3: Complements and Conditional Probability, and Bayes' Theorem
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.1: Basic Concepts of Probability
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 1: Introduction to Statistics
Section 1.3: Collecting Sample Data
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.1: Measures of Center
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 2: Exploring Data with Tables and Graphs
2.1: Frequency Distributions for Organizing and Summarizing Data
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 2: Exploring Data with Tables and Graphs
2.2: Histograms
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 2: Exploring Data with Tables and Graphs
2.3: Graphs that Enlighten and Graphs that Deceive
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.3: Complements and Conditional Probability, and Bayes' Theorem
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.1: Basic Concepts of Probability
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 1: Introduction to Statistics
Section 1.3: Collecting Sample Data
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.1: Measures of Center
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 2: Exploring Data with Tables and Graphs
2.4: Scatterplots, Correlation, and Regression
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.3: Sampling Distributions and Estimators
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.2: Measures of Variation
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 8: Hypothesis Testing
8.1: Basics of Hypothesis Testing
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 1: Introduction to Statistics
Section 1.2: Types of Data, Key Concept
Solution to the practice test ch 10 correlation reg ch 11 gof ch12 anovaLong Beach City College
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 10: Correlation and Regression
10.1: Correlation
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.5: Assessing Normality
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.2: Addition Rule and Multiplication Rule
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.2: Real Applications of Normal Distributions
Measures of Central Tendency, Variability and ShapesScholarsPoint1
The PPT describes the Measures of Central Tendency in detail such as Mean, Median, Mode, Percentile, Quartile, Arthemetic mean. Measures of Variability: Range, Mean Absolute deviation, Standard Deviation, Z-Score, Variance, Coefficient of Variance as well as Measures of Shape such as kurtosis and skewness in the grouped and normal data.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 2
Chapter 4: Probability
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 1
Module 1: Chapters 1-3
Chapter 1: Introduction to Statistics.
Chapter 2: Exploring Data with Tables and Graphs.
Chapter 3: Describing, Exploring, and Comparing Data.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 3
Practice Test Chapter 6 (Normal Probability Distributions)
Chapter 6: Normal Probability Distributions
Chapter 5 part2- Sampling Distributions for Counts and Proportions (Binomial ...nszakir
Mathematics, Statistics, Sampling Distributions for Counts and Proportions, Binomial Distributions for Sample Counts,
Binomial Distributions in Statistical Sampling, Binomial Mean and Standard Deviation, Sample Proportions, Normal Approximation for Counts and Proportions, Binomial Formula
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 2: Exploring Data with Tables and Graphs
2.4: Scatterplots, Correlation, and Regression
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.3: Sampling Distributions and Estimators
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 3: Describing, Exploring, and Comparing Data
3.2: Measures of Variation
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 8: Hypothesis Testing
8.1: Basics of Hypothesis Testing
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 1: Introduction to Statistics
Section 1.2: Types of Data, Key Concept
Solution to the practice test ch 10 correlation reg ch 11 gof ch12 anovaLong Beach City College
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 10: Correlation and Regression
10.1: Correlation
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.5: Assessing Normality
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.2: Addition Rule and Multiplication Rule
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.2: Real Applications of Normal Distributions
Measures of Central Tendency, Variability and ShapesScholarsPoint1
The PPT describes the Measures of Central Tendency in detail such as Mean, Median, Mode, Percentile, Quartile, Arthemetic mean. Measures of Variability: Range, Mean Absolute deviation, Standard Deviation, Z-Score, Variance, Coefficient of Variance as well as Measures of Shape such as kurtosis and skewness in the grouped and normal data.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 2
Chapter 4: Probability
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 1
Module 1: Chapters 1-3
Chapter 1: Introduction to Statistics.
Chapter 2: Exploring Data with Tables and Graphs.
Chapter 3: Describing, Exploring, and Comparing Data.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 3
Practice Test Chapter 6 (Normal Probability Distributions)
Chapter 6: Normal Probability Distributions
Chapter 5 part2- Sampling Distributions for Counts and Proportions (Binomial ...nszakir
Mathematics, Statistics, Sampling Distributions for Counts and Proportions, Binomial Distributions for Sample Counts,
Binomial Distributions in Statistical Sampling, Binomial Mean and Standard Deviation, Sample Proportions, Normal Approximation for Counts and Proportions, Binomial Formula
TSTD 6251 Fall 2014SPSS Exercise and Assignment 120 PointsI.docxnanamonkton
TSTD 6251 Fall 2014
SPSS Exercise and Assignment 1
20 Points
In this class, we are going to study descriptive summary statistics and learn how to construct box plot. We are still working with univariate variable for this exercise.
Practice Example:
Admission receipts (in million of dollars) for a recent season are given below for the
n =
30 major league baseball teams:
19.4 26.6 22.9 44.5 24.4 19.0 27.5 19.9 22.8 19.0 16.9 15.2 25.7 19.0 15.5 17.1 15.6 10.6 16.2 15.6 15.4 18.2 15.5 14.2 9.5 9.9
10.7 11.9 26.7 17.5
Require:
a. Compute the mean, variance and standard deviation.
b. Find the sample median, first quartile, and third quartile.
c. Construct a boxplot and interpret the distribution of the data.
d. Discuss the distribution of this set of data by examining kurtosis and skewness
statistics, such as if the distribution is skewed to one side of the distribution, and if the
distribution shows a peaked/skinny curve or a spread out/flat curve.
SPSS Procedures for Computing Summary Statistics
:
Enter the 30 data values in the first column of SPSS
Data View
Tab
Variable View
and name this variable
receipts
Adjust
Decimals
to 3 decimal points
Type
Admission Receipts
($ mn)
in the
Label
column for output viewer
Return to
Data View
and click
A
nalyze
on the menu bar
Click the second menu
D
e
scriptive Statistics
Click
F
requencies …
Move
Admission Receipts
to the
Variable(s)
list by clicking the arrow button
Click
S
tatistics …
button at the top of the dialog box
Now, you can select the descriptive statistics according to what the question requires. For this practice question, it requires central tendency, dispersion, percentile and distribution statistics, so we click all the boxes
except for
P
ercentile(s): and Va
l
ues are group midpoints
.
Click
Continue
to return to the
Frequencies
dialog box
Click
OK
to generate descriptive statistic output which is pasted below:
The first table provides summary statistics and the second table lists frequencies, relative frequencies and cumulative frequencies. The statistics required for solving this problem are highlighted in red.
Statistics
Admission Receipts
N
Valid
30
Missing
0
Mean
18.76333
Std. Error of Mean
1.278590
Median
17.30000
Mode
19.000
Std. Deviation
7.003127
Variance
49.043782
Skewness
1.734
Std. Error of Skewness
.427
Kurtosis
5.160
Std. Error of Kurtosis
.833
Range
35.000
Minimum
9.500
Maximum
44.500
Sum
562.900
Percentiles
10
10.61000
20
14.40000
25
15.35000
30
15.50000
40
15.84000
50
17.30000
60
19.00000
70
19.75000
75
22.82500
80
24.10000
90
26.69000
Admission Receipts
Frequency
Percent
Valid Percent
Cumulative Percent
Valid
9.500
1
3.3
3.3
3.3
9.900
1
3.3
3.3
6.7
10.600
1
3.3
3.3
10.0
10.700
1
3.3
3.3
13.3
11.900
1
3.3
3.3
16.7
14.200
1
3.3
3.3
20.0
15.2.
1. Outline the differences between Hoarding power and Encouraging..docxpaynetawnya
1. Outline the differences between Hoarding power and Encouraging.
2. Explain about the power of Congruency in Leadership.
DataIDSalaryCompaMidpoint AgePerformance RatingServiceGenderRaiseDegreeGender1GrCopy Employee Data set to this page.822.10.962233290915.81FAThe ongoing question that the weekly assignments will focus on is: Are males and females paid the same for equal work (under the Equal Pay Act)? 1522.60.984233280814.91FANote: to simplfy the analysis, we will assume that jobs within each grade comprise equal work.3522.60.984232390415.30FA37230.999232295216.20FAThe column labels in the table mean:1023.11.003233080714.71FAID – Employee sample number Salary – Salary in thousands 2323.11.004233665613.30FAAge – Age in yearsPerformance Rating – Appraisal rating (Employee evaluation score)1123.31.01223411001914.81FASERvice – Years of serviceGender: 0 = male, 1 = female 2623.51.020232295216.20FAMidpoint – salary grade midpoint Raise – percent of last raise3123.61.028232960413.91FAGrade – job/pay gradeDegree (0= BS\BA 1 = MS)3623.61.026232775314.30FAGender1 (Male or Female)Compa-ratio - salary divided by midpoint4023.81.034232490206.30MA14241.04523329012161FA4224.21.0512332100815.71FA1924.31.055233285104.61MA25251.0872341704040MA3226.50.855312595405.60MB227.70.895315280703.90MB3428.60.923312680204.91MB3933.91.094312790615.50FB2034.11.1013144701614.80FB1834.51.1133131801115.60FB335.11.132313075513.61FB1341.11.0274030100214.70FC741.31.0324032100815.71FC1642.21.054404490405.70MC4145.81.144402580504.30MC2746.91.172403580703.91MC548.21.0044836901605.71MD3049.31.0274845901804.30MD2456.31.173483075913.80FD4556.91.185483695815.21FD4757.21.003573795505.51ME3357.51.008573590905.51ME4581.01857421001605.51ME3858.81.0325745951104.50ME5059.61.0465738801204.60ME4660.21.0575739752003.91ME2260.31.257484865613.81FD161.61.081573485805.70ME4461.81.0855745901605.21ME49631.1055741952106.60ME1763.71.1185727553131FE1264.71.1355752952204.50ME4869.51.2195734901115.31FE973.91.103674910010041MF4375.61.1286742952015.50FF2976.31.139675295505.40MF2177.21.1526743951306.31MF678.11.1656736701204.51MF2878.31.169674495914.40FF
Week 2This assignment covers the material presented in weeks 1 and 2.Six QuestionsBefore starting this assignment, make sure the the assignment data from the Employee Salary Data Set file is copied over to this Assignment file.You can do this either by a copy and paste of all the columns or by opening the data file, right clicking on the Data tab, selecting Move or Copy, and copying the entire sheet to this file(Weekly Assignment Sheet or whatever you are calling your master assignment file).It is highly recommended that you copy the data columns (with labels) and paste them to the right so that whatever you do will not disrupt the original data values and relationships.To Ensure full credit for each question, you need to show how you got your results. For example, Question 1 asks for several data values. If you obtain them using descript ...
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 4
Chapter 9: Inferences about Two Samples
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 4
Chapter 8: Hypothesis Testing
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 4
Module 4:
Chapter 8, Hypothesis Testing
Chapter 9: Two Populations
Solution to the practice test ch 8 hypothesis testing ch 9 two populationsLong Beach City College
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 4
Module 4:
Chapter 8, Hypothesis Testing
Chapter 9: Two Populations
Solution to the Practice Test 3A, Chapter 6 Normal Probability DistributionLong Beach City College
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 3
Practice Test Chapter 6 (Normal Probability Distributions)
Chapter 6: Normal Probability Distributions
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 2 Solutions
Chapter 4: Probability
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 1
Module 1: Chapters 1-3
Chapter 1: Introduction to Statistics.
Chapter 2: Exploring Data with Tables and Graphs.
Chapter 3: Describing, Exploring, and Comparing Data.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 12: Analysis of Variance
12.2: Two-Way ANOVA
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 12: Analysis of Variance
12.1: One-Way ANOVA
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 11: Goodness-of-Fit and Contingency Tables
11.2: Contingency Tables
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 11: Goodness-of-Fit and Contingency Tables
11.1: Goodness of Fit Notation
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 10: Correlation and Regression
10.2: Regression
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 9: Inferences from Two Samples
9.4: Two Variances or Standard Deviations
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 9: Inferences from Two Samples
9.3 Two Means, Two Dependent Samples, Matched Pairs
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 9: Inferences from Two Samples
9.2: Two Means, Independent Samples
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2. Chapter 3:
Describing, Exploring, and Comparing Data
3.1 Measures of Center
3.2 Measures of Variation
3.3 Measures of Relative Standing and Boxplots
2
Objectives:
1. Summarize data, using measures of central tendency, such as the mean, median, mode,
and midrange.
2. Describe data, using measures of variation, such as the range, variance, and standard
deviation.
3. Identify the position of a data value in a data set, using various measures of position,
such as percentiles, deciles, and quartiles.
4. Use the techniques of exploratory data analysis, including boxplots and five-number
summaries, to discover various aspects of data
3. z Scores
A z score (or standard score or standardized value) is the number of standard
deviations that a given value x is above or below the mean. It is obtained by subtracting
the mean from the value and dividing the result by the standard deviation.
Sample: 𝑧 =
𝑥−𝑥
𝑠
Population:𝑧 =
𝑥−𝜇
𝜎
3.3 Measures of Relative Standing and Boxplots
Values: z scores ≤ −2.00 or z scores ≥ 2.00 3
Important Properties of z Scores
1. A z score is the number of standard deviations that a given value x is above or below the mean.
2. z scores are expressed as numbers with no units of measurement.
3. A data value is significantly low if its z score is less than or equal to −2 or the value is
significantly high if its z score is greater than or equal to +2.
4. If an individual data value is less than the mean, its corresponding z score is a negative number.
4. Example 1
4
Which of the following two data values is more extreme relative to the data
set from which it came?
=
4000 − 3152
693.4
= 1.22
=
99 − 98.20
0.62
= 1.29
Temperature
The 4000 g weight of a baby (n = 400, 𝑥 = 3152.0𝑔, 𝑠 = 693.4𝑔)
The 990
𝐹 temperature of an adult (n = 106, 𝑥 = 98.200
𝐹, 𝑠 = 0.620
𝐹)
𝑧 =
𝑥 − 𝑥
𝑠
𝑧 =
𝑥−𝑥
𝑠
, 𝑧 =
𝑥−𝜇
𝜎
5. 5
A student scored 65 on a calculus test that had a mean of 50 and a standard
deviation of 10; she scored 30 on a history test with a mean of 25 and a
standard deviation of 5. Compare her relative positions on the two tests.
She has a higher relative position in the Calculus class.
History: 𝑧 =
30 − 25
5
Calculus: 𝑧 =
65 − 50
10
= 1.5
= 1.0
𝑧 =
𝑥 − 𝜇
𝜎
Example 2 𝑧 =
𝑥−𝑥
𝑠
, 𝑧 =
𝑥−𝜇
𝜎
6. 6
=
75 − 239.4
64.2
= −2.56
Interpretation: 𝑧 = −2.56 < −2 → the platelet count of 75 is significantly low.
Platelets clump together and form clots to stop the bleeding during injury.
The lowest platelet count in a dataset is 75 (platelet counts are measured in
1000 cells/𝜇𝐿), is this significantly low? (𝑥 = 239.4, 𝑠 = 64.2)
𝑧 =
𝑥 − 𝑥
𝑠
Example 3 𝑧 =
𝑥−𝑥
𝑠
, 𝑧 =
𝑥−𝜇
𝜎
7. Percentiles are measures of location, denoted P1, P2, . . . , P99, which divide a
set of data into 100 groups with about 1% of the values in each group.
(Percentiles separate the data set into 100 equal groups.)
A percentile rank for a datum represents the percentage of data values
below the datum (round the result to the nearest whole number):
3.3 Measures of Relative Standing and Boxplots
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 =
# of values below 𝑋
total # of values
⋅ 100% Some Texts: 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 =
# of values below 𝑋 + 0.5
total # of values
⋅ 100%
n total number of values in the data set
k percentile being used (Example: For the 25th percentile, k = 25.)
L locator that gives the position of a value (Example: For the 12th
value in the sorted list, L = 12.)
Pk kth percentile (Example: P25 is the 25th percentile.)
7
𝐿 =
𝑘
100
⋅ 𝑛 =
𝑃
100
⋅ 𝑛
8. 8
Fifty (50) cell phone data speeds listed below are arranged in increasing order.
Find the percentile for the data speed of 11.8 Mbps.
0.8 1.4 1.8 1.9 3.2 3.6 4.5 4.5 4.6 6.2
6.5 7.7 7.9 9.9 10.2 10.3 10.9 11.1 11.1 11.6
11.8 12.0 13.1 13.5 13.7 14.1 14.2 14.7 15.0 15.1
15.5 15.8 16.0 17.5 18.2 20.2 21.1 21.5 22.2 22.4
23.1 24.5 25.7 28.5 34.6 38.5 43.0 55.6 71.3 77.8
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 =
# of values below 11.8
total # of values
⋅ 100% =
20
50
⋅ 100% = 40%
Interpretation:
A data speed of 11.8 Mbps is in the 40th percentile and separates the lowest
40% of values from the highest 60% of values. We have P40 = 11.8 Mbps.
Example 4 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 =
# of values below 𝑋
total # of values
⋅ 100%, 𝐿 =
𝑘
100
⋅ 𝑛 =
𝑃
100
⋅ 𝑛
9. Example 5
9
For the 50 cell phone data speeds listed, a) find the
20th percentile (P20), and b) the 87th percentile (P87).
0.8 1.4 1.8 1.9 3.2 3.6 4.5 4.5 4.6 6.2
6.5 7.7 7.9 9.9 10.2 10.3 10.9 11.1 11.1 11.6
11.8 12.0 13.1 13.5 13.7 14.1 14.2 14.7 15.0 15.1
15.5 15.8 16.0 17.5 18.2 20.2 21.1 21.5 22.2 22.4
23.1 24.5 25.7 28.5 34.6 38.5 43.0 55.6 71.3 77.8
Converting a Percentile to a Data Value
𝐿 =
𝑘
100
⋅ 𝑛 =
𝑃
100
⋅ 𝑛
Solution: k = 20, n = 50
=
20
100
⋅ 50
Whole Number: The value of the 20th
percentile is between the Lth (10th ) value and
the L + 1st (11st )value.
The 20th percentile: 𝑃20 =
6.2+6.5
2
= 6.35𝑀𝑏𝑝𝑠
Procedure:
1. Rank the data (Lo to Hi).
2. Find Index or locator: 𝐿 =
𝑘
100
⋅ 𝑛 (k
= % & n = sample size).
3. 𝐿 is not a whole number, round it up
to the next whole number, the Lth
value is the kth percentile.
4. 𝐿 is a whole number, the kth percentile
is the average of that value and the next
one in your data.
𝐿 =
𝑘
100
⋅ 𝑛 =
𝑃
100
⋅ 𝑛 = 10
Solution: k = 87, n = 50
𝐿 =
87
100
⋅ 50 = 43.5
Not a Whole Number: Round
up ALWAYS: The value of the
87th percentile is the 44th value.
𝑃87 = 28.5 𝑀𝑏𝑝𝑠
10. Quartiles
Quartiles are measures of location, denoted Q1, Q2, and Q3, which divide a set of data into four groups with
about 25% of the values in each group. (Quartiles separate the data set into 4 equal groups. Q1=P25, Q2=MD,
Q3=P75 )
Q1 (First quartile, or P25) It separates the bottom 25% of the sorted values from the top 75%.
Q2 (Second quartile, or P50 ) and same as the median. It separates the bottom 50% of the sorted values from the top
50%.
Q3 (Third quartile): Same as P75. It separates the bottom 75% of the sorted values from the top 25%.
Caution Just as there is not universal agreement on a procedure for finding percentiles, there is not universal
agreement on a single procedure for calculating quartiles, and different technologies often yield different results.
Deciles separate the data set into 10 equal groups. D1=P10, D4=P40
The Interquartile Range, IQR = Q3 – Q1.
Step 1 Arrange the data in order from lowest to highest.
Step 2 Find the median of the data values. This is the value for Q2.
Step 3 Find the median of the data values that fall below Q2. This is the value for Q1.
Step 4 Find the median of the data values that fall above Q2. This is the value for Q3.
10
3.3 Measures of Relative Standing and Boxplots
𝐼𝑄𝑅 = 𝑄3 − 𝑄1
𝑆𝑒𝑚𝑖 𝐼𝑄𝑅 = (𝑄3−𝑄1)/2
𝑀𝑖𝑑 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝑟𝑎𝑛𝑔𝑒: = (𝑄3+𝑄1)/2
10-90 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝑟𝑎𝑛𝑔𝑒: = 𝑃90 − 𝑃10
11. Example 7
11
Given the fifty (50) data speeds listed, find the
5-number summary.
0.8 1.4 1.8 1.9 3.2 3.6 4.5 4.5 4.6 6.2
6.5 7.7 7.9 9.9 10.2 10.3 10.9 11.1 11.1 11.6
11.8 12.0 13.1 13.5 13.7 14.1 14.2 14.7 15.0 15.1
15.5 15.8 16.0 17.5 18.2 20.2 21.1 21.5 22.2 22.4
23.1 24.5 25.7 28.5 34.6 38.5 43.0 55.6 71.3 77.8
5-Number Summary
Min = 0.8 Mbps & Max = 77.8 Mbps
The median is equal to MD = Q2 = (13.7 + 14.1) / 2 = 13.9 Mbps.
5-Number Summary
Consists of:
1. Minimum
2. First quartile, Q1
3. Second quartile, Q2
(same as the median)
4. Third quartile, Q3
5. Maximum
Q1 = 7.9 Mbps (Median of the bottom half)
Q3 = 21.5 Mbps (Median of the top half)
12. Example 8
12
Given the fifty (50) data speeds listed, construct a
boxplot. Identify Outliers if any.
0.8 1.4 1.8 1.9 3.2 3.6 4.5 4.5 4.6 6.2
6.5 7.7 7.9 9.9 10.2 10.3 10.9 11.1 11.1 11.6
11.8 12.0 13.1 13.5 13.7 14.1 14.2 14.7 15.0 15.1
15.5 15.8 16.0 17.5 18.2 20.2 21.1 21.5 22.2 22.4
23.1 24.5 25.7 28.5 34.6 38.5 43.0 55.6 71.3 77.8
Boxplots A boxplot (or box-and-
whisker diagram) is a
graph that consists of a
line extending from the
min to the max value,
and a box with lines
drawn at the first
quartile Q1, the median,
and the third quartile Q3.
Min = 0.8, Q1 = 7.9, Q2 = 13.9, Q3 = 21.5, Max = 77.8
Constructing a Boxplot
1. Find the 5-number
summary (Min, Q1, Q2,
Q3, Max).
2. Construct a line segment
extending from the
minimum to the maximum
data value.
3. Construct a box
(rectangle) extending from
Q1 to Q3, and draw a line
in the box at the value of
Q2 (median).
𝐼𝑄𝑅 = 𝑄3 − 𝑄1 = 21.5 − 7.9 = 13.6, 𝑄1 − 1.5𝐼𝑄𝑅 = −12.5 & 𝑄3 + 1.5𝐼𝑄𝑅 = 41.9
Any numbers smaller than −12.5 and larger than 41.9 is considered an outlier.
43.0, 55.6, 71.3 & 77.8
13. Skewness
A boxplot can often be used to identify skewness. A distribution of data is skewed if it
is not symmetric and extends more to one side than to the other.
Modified Boxplots
A modified boxplot is a regular boxplot constructed with these modifications:
1. A special symbol (such as an asterisk or point) is used to identify outliers as defined
above, and
2. the solid horizontal line extends only as far as the minimum data value that is not an
outlier and the maximum data value that is not an outlier.
Identifying Outliers (An outlier is a value that lies very far away from the vast majority of
the other values in a data set.) for Modified Boxplots
1. Find the quartiles Q1, Q2, and Q3.
2. Find the interquartile range (IQR), where IQR = Q3 − Q1.
3. Evaluate 1.5 × IQR.
4. In a modified boxplot, a data value is an outlier if it is above Q3, by an amount greater
than 1.5 × IQR or below Q1, by an amount greater than 1.5 × IQR.
13
3.3 Measures of Relative Standing and Boxplots Boxplots
14. Example 9
14
Given the fifty (50) data speeds listed, find the
40th percentile, denoted by P40.
0.8 1.4 1.8 1.9 3.2 3.6 4.5 4.5 4.6 6.2
6.5 7.7 7.9 9.9 10.2 10.3 10.9 11.1 11.1 11.6
11.8 12.0 13.1 13.5 13.7 14.1 14.2 14.7 15.0 15.1
15.5 15.8 16.0 17.5 18.2 20.2 21.1 21.5 22.2 22.4
23.1 24.5 25.7 28.5 34.6 38.5 43.0 55.6 71.3 77.8
Converting a Percentile to a Data Value (Time)
Whole Number: The value of the 40th percentile is
between the Lth (20th ) value and the 21st value.
The 40th percentile is P40 = 11.7 Mbps.
𝐿 =
𝑘
100
⋅ 𝑛 =
𝑃
100
⋅ 𝑛
Procedure:
1. Rank the data (Lo to Hi).
2. Find Index or locator: 𝐿 =
𝑘
100
⋅ 𝑛 (k
= % & n = sample size).
3. 𝐿 is not a whole number, round it up
to the next whole number, the Lth
value is the kth percentile.
4. 𝐿 is a whole number, the kth percentile
is the average of that value and the next
one in your data.
Solution: k = 40, n = 50
=
40
100
⋅ 50
𝐿 =
𝑘
100
⋅ 𝑛 =
𝑃
100
⋅ 𝑛 = 20
15. Example 10
15
A teacher gives a 20-point test to 10 students. a) Find the percentile rank of a score of
12. b) Find the value corresponding to the 25th percentile.
18, 15, 12, 6, 8, 2, 3, 5, 20, 10
100
n k
L
Sort in ascending order.
2, 3, 5, 6, 8, 10, 12, 15, 18, 20
Interpretation:
A student whose score
was 12 did better than
65% of the class.
6 values
=
6 + 0.5
10
⋅ 100% = 65%
Some Texts: 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 =
# of values below 𝑋 + 0.5
total # of values
⋅ 100%
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 =
# of values below 𝑋 + 0.5
total # of values
⋅ 100%
b) Sort in ascending order.
2, 3, 5, 6, 8, 10, 12, 15, 18, 20
𝑐 =
𝑛 ⋅ 𝑝
100
=
10 ⋅ 25
100
= 2.5 ≈ 3
Interpretation:
The value 5 corresponds
to the 25th percentile.