Math 365, Elementary Statistics 

Lesson 2 : Measures of Central Tendency and Measures of DispersionIntroductionIn this lesson we talk about two types of constants that we compute from data:
A measure of central tendency represents an "average value." Mean, median, mode (if you already know these) are measures of central tendency. A measure of dispersion is a measure of how widely the data is scattered around. 2.1 Measure of Central Tendency: MeanThe most common measure of central tendencies is the mean or arithmetic mean. Definition. The mean
or the arithmetic mean of a set of data is
given by
If we denote a data value (i.e., the variable) by x and if n is the
size of the data, then the above formula is written as
OR mean = x = ∑ x/n where ∑ denotes summation. If the data is a sample, then the mean is called the sample
mean. Again, if x denotes the variable, the data is sometimes
denoted by x_{1},x_{2}, ... ,x_{n} and then
OR
If you have not seen the notation ∑ before,
it simply means summation. For example,
Sometimes, different values in data carry different weight. Let us consider the following data and the corresponding frequency distribution that we computed earlier: Example 2.1.1 To estimate the mean time taken to complete a threemile drive by a race car, the race car did several time trials. The following are sample times taken (in seconds) to complete the laps:
Following is the frequency distribution of this data:
Now we want to compute the mean time. So, we add all the data values
and divide by the data size 35. We already have computed the frequency
distribution which tells us that, in the data, 46 was present 1 time,
47 was present 1 time, 48 was present 3, times and so on. So, using
the frequency distribution, we compute the mean as follows :
The mean of the original data is the weighted
mean of the data values 46, 47, 48, 49, 50, 51, 52, 53, 54, 55
and 56 with the corresponding frequency as the weight.
So, a new formula for the mean would be
OR
where f_{i} is the frequency of x_{i}. The weighted mean is defined in more general context as follows: Definition. If x_{1}, x_{2},
... , x_{n} in a data set have different weights and the values
x_{i} has weight w_{i}, then the weighted
mean is defined as
weighted mean = x = ∑w_{i}x_{i}
/ ∑w_{i}
Properties of MeanRemark (effect of translation): Your teacher tells you that the mean score for the midterm in your class is 73. After you complained and requested a change, he agreed that all can add 7 points to their score. The new mean score is (old mean + 7) = 73 + 7 = 80. This is what we meant by "effect of translation." Example (effect of multiplication by c): Suppose you have some data x_{1}, x_{2}, ..., x_{n} on salaries in an industry in the United States and the mean is $37000. On a certain day, 1 U.S. dollar = 1.4729 Canadian dollars (say c = 1.4729). So, in Canadian dollars the mean is 37000*c = 37000 x 1.4729. Similarly, the change of units (inches to feet or cm) are "multiplication by a constant c." Example 2.1.2. A student took PHSX 115 (College Physics), PSYC 120 (Personality), FREN 110 (Elementary French), BUS 241 (Managerial Accounting), and MATH 365 (Elementary Statistics). The number of credit hours and the student's grade is given in the following table:
What is the student's GPA? Solution. The GPA is the weighted average of the points (corresponding to the grades), weight being the coursecredit hours. So, the GPA = (3x4+4x3+3x5+2x3+3x3)/(4+3+5+3+3) = 54/18 = 3. 2.2 Measure of Central Tendency: Median, and Mode
The Median

Use of Calculators (TI83): 

Entering your data

Sorting data and
computing the median

Computing the mean
if only raw data is given

Computing the mean
if the frequency table is given

Problems on 2.2: Mean and Median
Exercise 2.2.1. The following is the price
(in dollars) of a stock (say, CISCO SYSTEMS) checked by a trader several
times on a particular day.
138  142  127  137  148  130  142  133 
Find the median price and mean price observed by the trader.
Solution
Exercise 2.2.2. The following figures refer to the GPA of six
students.
3.0  3.3  3.1  3.0  3.1  3.1 
Find the median and mean GPA.
Exercise 2.2.3. The following data give the lifetime (in days) of light bulbs.
138  952  980  967  992  197  215  157 
Find the mean and median lifetime of these bulbs.
Solution
Exercise 2.2.4. An athlete ran an event 32
times. The following frequency table gives the time taken (in seconds)
by the athlete to complete the events.
Time (in seconds)  Frequency 

26  3 
27  6 
28  5 
29  6 
30  9 
31  3 
Total  32 
Compute the mean and median time taken by the athlete.
Solution
Exercise 2.2.5. Following is data on the weight (in ounces), at birth, of 96 babies born in Lawrence Memorial Hospital in May 2000.
94  105  124  110  119  137  96  110  120  115  119 
104  135  123  129  72  121  117  96  107  80  80 
96  123  124  124  134  78  138  106  130  97  134 
111  133  128  96  126  124  125  127  62  127  96 
116  118  126  94  127  121  117  124  93  135  112 
120  125  120  147  138  72  119  89  81  113  100 
109  127  138  122  110  113  100  115  110  135  120 
97  127  120  110  107  111  126  132  120  108  148 
133  103  92  124  150  86  121  98 
Compute the mean and median weight, at birth, of the babies.
Solution
Exercise 2.2.6. Following is data on the hourly wages (paid only in whole dollars) of 99 employees in an industry.
7  11  7  11  10  9  10  10  12  13 
7  8  11  11  14  9  7  9  11  7 
9  13  12  14  7  8  7  14  15  9 
9  7  11  9  12  9  12  11  14  9 
12  13  7  9  10  14  11  12  13  7 
15  15  16  16  15  16  11  7  18  19 
15  16  15  15  16  16  17  16  16  13 
15  15  16  15  16  15  15  17  16  12 
16  15  15  16  15  15  19  8  16  17 
16  16  15  16  16  16  13  12  8 
Compute the mean and median hourly wage.
Solution
Exercise 2.2.7. Following is the frequency table on the number of typos in a sample of 30 books published by a publisher.
No. of Typos  156  158  159  160  162 

Frequency  6  4  5  6  9 
Find the mean and median number of typos in a book.
Solution
Exercise 2.2.8. Following is data on the length (in inches), at birth, of 96 babies born in Lawrence Memorial Hospital in May 2000.
18  18.5  19  18.5  19  21  18  19  20  20.5 
19  19  21.5  19.5  20  17  20  20  19  20.5 
18  18.5  20  19.5  20.75  20  21  18  20.5  20 
21  19  20.5  19  20  19.5  17.75  20  19.5  20 
20.5  17  21  18.5  20  20  20  18.5  19.5  19 
18  20.5  18  20  19  19  19.5  20  20.75  21 
17.75  19  18  19  20  18.5  20  19  21  19 
19.5  20  20  19  19.5  20  19.5  18.5  20.5  19.5 
20.25  20  19.5  19.5  20  20  20  21  20  19 
18.5  20.5  21.5  18  19.5  18 
Compute the mean and median length, at birth, of these babies.
Solution
Range
Clearly, the measures of central tendency—mean, median, mode—cannot
tell us the "whole story" about the data.
Example 2.3.1. Suppose two sections of the statistics class have the following percentage score distribution at the end of the semester:
Section A  81  84  83  80  82 

Section B  72  93  92  82  71 
Both these sections have the same mean—82. But in Section A, everybody will get a B grade. In section B, we will have two C's, one B and two A's.
The measure of dispersion is a measure of how widely the data is scattered around. In section A, the data has a very small dispersion or variability, whereas section B has a large dispersion.
A very simple measure of dispersion is the range of the data as we have defined before:
range = largest value  smallest value.
Mean Deviation, Sample Variance, and Standard Deviation
We will discuss three more measures of dispersion.
Suppose we have a data set x_{1}, x_{2}, ... , x_{n} of size n. We will denote the mean of the data by x. Three definitions follow:
Definition. The mean deviation of the data is defined as follows.
mean deviation = ( x_{1} x
 + ... + x_{n} x
) / n
So, the mean deviation is the mean of the absolute deviations  x_{i} x  from the mean.
Definition. The sample
variance s^{2} of the data is
defined as follows:
s^{2} = ( (x_{1} x)^{2}
+ ... + (x_{n} x)^{2}
) / (n 1)
Remark.
Definition. The sample
standard deviation s is defined as the square root of the sample
variance s^{2}. So, to compute the sample standard deviation,
we have to compute the sample variance first.
If we simplify the definition of sample variance we get the following
formula:
s^{2} =( (x_{1}^{2} + x_{2}^{2}
+ ... + x_{n}^{2})  nx^{2})/(n
 1)
Let us quickly do some computation with the above example 2.3.1.
The mean deviation for section A = (1+2+1+2+0)/5= 6/5 and the mean deviation for section B = (10+11+10+0+11)/5= 42/5. Since the variability of section B was much higher, the mean deviation was very high.
Let us compute the the sample variances :
For section A the sample variance is
( (8182)^{2}+(8482)^{2}+(8382)^{2}+(8082)^{2}+(8282)^{2}
)/(51) =
(1+4+1+4+0) /4= 10/4 = 2.5 .
For section B the sample variance is
( (7282)^{2}+(9382)^{2}+(9282)^{2}+(8282)^{2}+(7182)^{2}
)/(51) =
(100+121+100+0+121) /4= 442/4.
Application of Standard deviation
The mean and the standard deviation tell us a lot about how the data is distributed.
Chebyshev's Rule. This rule applies for all
kinds of data. Suppose x is the mean and s is
the standard deviation of the data. Then we have the following:
Chebyshev's Rule makes no assumption about the data or the variable. If we make some assumptions about the data, then we can improve the above rule as follows.
The Empirical Rule: Suppose the histogram
of the data is symmetric around the vertical line x = x
as follows:
In other words, the histogram should fit into a bellshaped curve.
Bellshaped Curve 
Click to see the Flash
animation.
Then we have the following:
Question: What does it mean when the variance or mean deviation of some data is zero? The answer is that all the data members are EQUAL!
Practice Problem. Consider the exercises 2.2.1 through 2.2.8. For each problem, compute the mean and standard deviation of the data and find what percentage of the data are within one, two, or three standard deviations from the mean.
Use of the Frequency Table
When a frequency table is given, we can use new formulas to compute the mean and variance of the data.
Formulas. Suppose the data consisting of n
observations are given in a frequency table (ungrouped). Let x_{i}
denote the values and f_{i} be the frequency of x_{i}.
Then
x 
= 

= 
n 
, 
s^{2} = 
n 1 
, 
s^{2} =  1
n 1 
[∑ 
(f_{i}x_{i}^{2}) 
n x^{2} 
]. 
Example 2.3.2. The following table extends the frequency table of the time taken to complete a lap by a race car (example 2.1.1) to compute mean and variance using the above formulas.
Time x 
Frequency f 
fx  fx^{2} 

46  1  46  2116 
47  1  47  2209 
48  3  144  6912 
49  3  147  7203 
50  4  200  10000 
51  6  306  15606 
52  4  208  10816 
53  5  265  14045 
54  5  270  14580 
55  2  110  6050 
56  1  56  3136 
Total  35  1799  92673 
So, the mean x = 1799/35 =51.4 and variance
s^{2} = (92673  35x 51.4^{2})/(351) = 6.0118.
Example 2.3.3. Following is the class frequency distribution of the data on birth weight of some babies (exercise 1.2, Lesson 1):
Classes  Frequency f 
Class Mark x 
fx  fx^{2} 

60.580.5  9  70.5  634.5  44732.25 
80.5100.5  20  90.5  1810  163805 
100.5120.5  25  110.5  2762.5  305256.25 
120.5140.5  37  130.5  4828.5  630119.25 
140.5160.5  8  150.5  1204  181202 
Total  99  11239.5  1325114.75 
We can use the above formula to compute (approximate) variance and the standard deviation of the birth weight.
So, the mean x = 11239.5/99 = 113.53 and variance
s^{2} = (1325114.75  99 x 113.53^{2})/(991) = 500.997.
Remarks.
Comment: We have had detailed discussions of various formulas for defining the mean, variance, and other constants. It is important to understand these concepts and formulas.
It is equally important to appreciate the value and necessity of using
calculators or other available software (like Excel). It is almost impossible
(and unnecessary) to compute these constants manually and correctly, unless
one is specially gifted with numerical computations.
Use of Calculators (TI83): 

Computing the variance
and standard deviation

Problems on 2.3: Variance, Standard Deviation, and Use of the Frequency Table
Exercise 2.3.1. The following is the price (in dollars) of a stock (say, CISCO SYSTEMS) checked by a trader several times on a particular day.
138  142  127  137  148  130  142  133 
Find the variance and standard deviation of the price.
Solution
Exercise 2.3.2. The following figures refer to the GPA of six students.
3.0  3.3  3.1  3.0  3.1  3.1 
Find the variance and standard deviation of GPA.
Exercise 2.3.3. The following data give the lifetime (in days) of certain light bulbs.
138  952  980  967  992  197  215  157 
Find the variance and standard deviation of the lifetime of these
bulbs.
Solution
Exercise 2.3.4. An athlete ran an event 32 times. The following frequency table gives the time taken (in seconds) by the athlete to complete the events.
Time (in seconds)  Frequency 

15.6  3 
15.7  6 
15.8  5 
15.9  6 
16.0  9 
16.1  3 
Total  32 
Compute the variance and standard deviation of time taken by the athlete.
Solution
Exercise 2.3.5. Following is data on the weight (in ounces), at birth, of 96 babies born in Lawrence Memorial Hospital in May 2000.
94  105  124  110  119  137  96  110  120  115  119 
104  135  123  129  72  121  117  96  107  80  80 
96  123  124  124  134  78  138  106  130  97  134 
111  133  128  96  126  124  125  127  62  127  96 
116  118  126  94  127  121  117  124  93  135  112 
120  125  120  147  138  72  119  89  81  113  100 
109  127  138  122  110  113  100  115  110  135  120 
97  127  120  110  107  111  126  132  120  108  148 
133  103  92  124  150  86  121  98 
Compute the variance and standard deviation of the weight, at birth,
of these babies.
Solution
Exercise 2.3.6. Following is data on the hourly wages (paid only in whole dollars) of 99 employees in an industry.
7  11  7  11  10  9  10  10  12  13 
7  8  11  11  14  9  7  9  11  7 
9  13  12  14  7  8  7  14  15  9 
9  7  11  9  12  9  12  11  14  9 
12  13  7  9  10  14  11  12  13  7 
15  15  16  16  15  16  11  7  18  19 
15  16  15  15  16  16  17  16  16  13 
15  15  16  15  16  15  15  17  16  12 
16  15  15  16  15  15  19  8  16  17 
16  16  15  16  16  16  13  12  8 
Compute the variance and standard deviation of the hourly wages.
Solution
Exercise 2.3.7. Following is the frequency table on the number of typos in a sample of 30 books published by a publisher.
No. of Typos  156  158  159  160  162 

Frequency  6  4  5  6  9 
Find the mean number, variance, and standard deviation of typos in
a book.
Solution
Exercise 2.3.8. Following is data on the length (in inches), at birth, of 96 babies born in Lawrence Memorial Hospital in May 2000.
18  18.5  19  18.5  19  21  18  19  20  20.5 
19  19  21.5  19.5  20  17  20  20  19  20.5 
18  18.5  20  19.5  20.75  20  21  18  20.5  20 
21  19  20.5  19  20  19.5  17.75  20  19.5  20 
20.5  17  21  18.5  20  20  20  18.5  19.5  19 
18  20.5  18  20  19  19  19.5  20  20.75  21 
17.75  19  18  19  20  18.5  20  19  21  19 
19.5  20  20  19  19.5  20  19.5  18.5  20.5  19.5 
20.25  20  19.5  19.5  20  20  20  21  20  19 
18.5  20.5  21.5  18  19.5  18 
Compute the variance and standard deviation of the length, at birth,
of these babies.
Solution
Exercise 2.3.9. The following is the frequency table of weight (in pounds) of some salmon in a river. Find the variance and standard deviation.
Weight x  31  32  33  34  35  36  37 

Frequency f  3  2  4  5  6  5  9 
Find the variance and the standard deviation.
Solution
Exercise 2.3.10. The following data represents the time (in minutes) taken by students to drive to campus.
23  17  19  24  42  33  20  22  15  9 
26  37  29  19  35  18  30  21  11  23 
13  27  32  32  23  35  25  33  24  23 
Find the mean, variance, and the standard deviation of the data.
Solution