Data Setting: A graduate student in Psychology was asked to grade 40 final exams, selected at
random from several large sections of an introductory course. The resulting scores are found
below.
77
84
91
50
75
68
92
74
81
92
86
96
61
37
80
84
83
52
60
75
95
62
83
85
78
98
83
73
100
71
87
81
85
79
64
71
85
78
81
65
The professor for the class asked her how the students performed?
Unfortunately, the graduate student found that looking at the list of scores was about as
informative as looking at a scrambled set of letters.
To get information out of the data, she needed to summarize the data.
So, how should she go about summarizing the scores on the test?
Measures of Central Tendency
Measures of central tendency, or more simply measures of center, indicate where the center or
most typical value of a data set lies.
Three common measures of central tendency are the arithmetic mean (mean), median, and mode.
The mean is the sum of the observations divided by the number of observations.
We use
x
to denote sample mean and  to denote population mean.
Example The following are salaries for four randomly selected employees at Microsoft.
$30,000
x
$35,000
$32,500
$36,000
30,000  35,000  32,500  36,000
 $33,375
4
The median of a set of observations is defined to be the middle value when the observations are
arranged from lowest to highest.

If the number of observations is odd, then the median is the observation exactly in the
middle of the data set.

If the number of observations is even, then the median is the mean of the two middle
observations in the ordered list.
Example Find the median for the sample of four salaries from Microsoft.
First put the salaries in order:
$30,000
$32,500
$35,000
$36,000
Median = 32,500  35,000
2

$33,750
The mode is the value that occurs most frequently in a data set.


There can be more than one mode.
When no value is repeated, we say there is no mode.
Example Find the mode for the following test scores
84
89
82
91
86
84
The mode is 84.
Which measure of central tendency do we use for a given data set?




For qualitative data, the mode is the only one of the three that makes sense.
For quantitative data, mean or median is often preferred over the mode as a measure of
center because the value that occurs most frequently may not necessarily be located near
the center of the data set.
If there are outliers in the data, median is preferred over the mean as the mean can be
misleading when outliers are present.
If there are no outliers, the mean is often the preferred choice because it has some nice
properties, which will be important in making inferences later in this course.
A weighted mean is the mean of a data set whose entries have varying weights. A weighted mean
is given by
x
 ( x  w)
w
where w is the weight of each entry x.
Example In this class, your grade is determined from the following sources: 20% from exam 1,
20% from exam 2, 25% from the final exam, 15% from lab, 15% from homework, and 5% from
the project. If you end up with the following scores, what is your final average in this class?
Source
Exam 1
Exam 2
Final Exam
Lab
Homework
Project
Score, x
84
75
72
90
100
85
x
Weight, w
.2
.2
.25
.15
.15
.05
 w =1
 ( x  w) 82.55  82.55
w = 1
xw
16.8
15
18
13.5
15
4.25
 ( x  w) =82.55
Measures of Variability
Measures of central tendency provide only a partial description of a quantitative data set. The
description is incomplete without a measure of the variability, or spread, of the data set.
For example, consider the following three data sets, which contain tests scores for students in
three different classes.
C1
C2
C3
50
50
70
Test Scores
60
70
73
74
73
74
80
76
76
Mean Median
90
100
77
100
77
80
75
75
75
75
75
75
The three data sets all have the same center but where they differ is in terms of variation in the
scores.
Three common measures of variability are the range, inter-quartile range, variance, and standard
deviation.
The range of a data set is equal to the largest measurement minus the smallest measurement.
Example Calculate the range for class 1 scores.
Range = 100 – 50 = 50
Although the range is easy to compute, it is sensitive to outliers.
The inter-quartile range (IQR) of a set of measurements is defined to be the difference between
the 75th percentile and the 25th; that is,
IQR=75th percentile(third quartile) – 25th percentile(first quartile)
To calculate the first and third quartiles
1. Arrange the observations in increasing order and locate the median in the ordered list of
observations.
2. The first quartile is the median of the observations whose position in the ordered list is to
the left of the location of the overall median.
3. The third quartile is the median of the observations whose position in the ordered list is to
the right of the location of the overall median.
Here are examples that show how the rules for the quartiles work for both odd and even numbers
of observations.
Example Finding the quartiles: odd number of observations.
Here are the travel times in minutes for 15 workers in North Carolina, chosen at random by the
Census Bureau:
30
20
10
40
25
20
10
60
15
40
5
30
12
10
10
Order the data : 5 10 10 10 10 12 15 20 20 25 30 30 40 40 60
Median=20
First quartile = 10
Third Quartile= 30
Example Finding the quartiles: even number of observations.
Here are the travel times to work for 20 New York Workers.
5 10 10 15 15 15 15 20 20
20 | 25 30 30 40 40 45 60
60 65 85
Location of the median is denoted by the vertical bar.
Median = 22.5
First quartile = 15
Third quartile = 42.5
The IQR does not provide a lot of useful information about the variability of a single set of
measurements, but it can be quite helpful in comparing the variability of two or more data sets.
Unlike the range, the IQR is not sensitive to outliers.
The sample variance for a sample of n measurements is equal to the sum of the squared distances
from the mean divided by (n - 1). In symbols, using s2 to represent the sample variance,
s2 =  ( x  x )
2
n 1
In calculating s2, we divide by (n – 1) instead of n to get a better estimate of the population
variance Using n in the formula for s2, we tend to underestimate 
The sample standard deviation, s, is defined as the positive square root of the sample variance, s2.
Example Calculate s2 and s for the class 1 scores
x
xx
50
60
70
80
90
100
-25
-15
-5
5
15
25
 (x  x)  0
s2 =  ( x  x )
n 1
s=
2

( x  x )2
625
225
25
25
225
625
2
 (x  x)
 1,750
1750
 350
5
350  18.708
Variance and Standard deviation are useful for comparing variability in two data sets. The data
set with the larger variance, or standard deviation, exhibits more variation in the data.
Standard deviation has the advantage over variance in that it provides a measure of variability in
the same units as the original data.
Thus, the standard deviation for a set of incomes would be in dollars whereas the variance would
be in dollars2.
Standard deviation and variance are sensitive to outliers.
Standard deviation can be used in conjunction with the mean to describe the variability in a
single data set.
Empirical Rule: For a distribution that is approximately bell-shaped,
Approximately:
 68% of the observations fall within one standard deviation of the mean.
 95% of the observations fall within two standard deviations of the mean.
 99.7% of the observations fall within three standard deviations of the mean.