Chapter 2 : Measures of Central Tendency and
Measures of Dispersion
Satya Mandal
Introduction
In this lesson we define various numerical measures
(constants) for
data sets. These numerical measures summarize and describe the data.
The average value of the data would be a common example.
There are two broad classification such numerical measures that are computed
from the data:
 measures of central tendency and
 measures of dispersion.
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: Mean
The 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
mean = 
sum of all the data values
size of the data

. 
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
mean = 
x

= 
n
∑ x_{i}
i=1
n

. 
OR
mean = x =

n
∑ x_{i}/n
i=1 
If you have not seen the notation ∑ before,
it simply means summation. For example,

n
∑
i = 1

x_{i} = x_{1}+x_{2}+
... +x_{n} 
Weighted Mean
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:
50 
48 
49 
46 
54 
53 
52 
51 
47 
56 
52 
51 
51 
53 
50 
49 
48 
54 
53 
51 
52 
54 
54 
53 
55 
48 
51 
50 
52 
49 
51 
53 
55 
54 
50 

Following is the frequency distribution of this data:
Time (in seconds) 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
Frequency 
1 
1 
3 
3 
4 
6 
4 
5 
5 
2 
1 
To compute the mean time of the original dat,
we obviously, add all the data values
and divide by the data size 35. The frequency
distribution 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 :
mean=x= 
(46x1+47x1+48x3+49x3+50x4+51x6+52x4+53x5+54x5+55x2+56x1)
(1+1+3+3+4+6+4+5+5+2+1)

=1799/35=51.4 
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 frequencies
as the weight.
Therefore, when we compute the mean using the frequency table,
the formula for the mean would be
x

= mean = 
^{n}
∑ x_{i} f_{i}
i=1
^{n}
∑f_{i}
_{i=1} 
OR
mean

= x
= 
^{ n
} ^{n}
∑ f_{i} x_{i }/
∑f_{i}
_{i=1} _{i=1}

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 = 
^{n}
∑
_{i=1}

w_{i}x_{i} 

^{n}
∑
_{i=1} 
w_{i} 

. 
OR
weighted mean = x = ∑w_{i}x_{i}
/ ∑w_{i}
Properties of the Mean
 Combining two means. Suppose we have two
sets of data. The mean of the first set is x,
and the size of the first set is m; the mean of the second set is y,
and size of the second set is n. The mean of the combined data is
Combined mean =
(m x +ny)/(m+n)
This is the weighted mean of x, y
with weight m,n respectively.
 Effect of translation. Let x
be the mean of x_{1}, x_{2}, ... , x_{n}. Then
the mean of y_{1} = x_{1}+d, y_{2} = x_{2}+d,
... ; y_{n} = x_{n}+d is given by
y = x+d
 Effect of multiplication by a constant.
Let x be the mean of x_{1}, ... ,
x_{n}. Then the mean of
z_{1} = cx_{1}, z_{2} = cx_{2},
... , z_{n} = cx_{n}
is given by
z = cx
Example (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 (March 21, 2011), 1 U.S. dollar = 0.976469
Canadian dollars (say c = 0.976469).
So, in Canadian dollars the mean is 37000*c = 37000 x 0.976469=36129.35 Canadian Dollars.
Similarly,
any change of units (inches to feet or cm, minutes to seconds)
are "multiplication by
a constant c."
Example 2.1.2.
(GPA is an exampled of weighted mean).
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:
Course 
PHSX 115

PSYC 120 
FREN 110 
BUS 241 
MATH 365 
Grade (Points) 
B (3 points) 
A (4 points) 
B (3 points) 
C (2 points) 
B (3 points) 
Credit Hours 
4 
3 
5 
3 
3 
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.
Other Measures of Central Tendency: Median, and Mode
The Median
The median represents the middle value of the data. Half the data will
be less than or equal to the median, and half the data will be greater
than or equal to the median. You are above the median American income
if half the American population is making less than you make.
Definition. Suppose the data is arranged in
an increasing order (i.e., in an array). If the size of data is ODD then
the median is the middle value. If it is
EVEN, then the median is the mean of the
middle two values.
The Percentiles
Definition. For a number p between 0 to 100,
the pth percentile x_{p} of the data is a number such that at
least p percent of the data members are below x_{p} and at least
(100  p) percent of the data members are above x_{p}.
 The 25th percentile is called the first quartile
Q_{1}.
 The median is the 50th percentile, also called the second
quartile Q_{2}.
 The 75th percentile is called the third quartile
Q_{3}.
The Mode
There is one other measure of central tendencies that should be mentioned.
Definition. The MODE
of the data is the value or values that have the highest frequency. For
example, the mode of the set {1, 3, 5, 5, 7} is {5} because it has the
highest frequency. The mode of {1, 1, 3, 5, 5, 7} is {1, 5} because 1
and 5 both have the highest frequency. Such a set is said to be bimodal.
Use of Calculators (TI84): 
Entering your data
 Press the button stat.
 Select "Edit" in the Edit menu and enter.
 You will find six lists named L1, L2, L3, L4, L5, L6.
 Let's say you want to enter your data in L1.
 If L1 has some data, clear it by pressing the stat button
and selecting ClrList in the Edit menu.
 Once L1 is cleared, select Edit in the Edit menu and enter.
 Now type in your data and enter one by one.

Sorting data and
computing the median
 Enter your data in a list, say L1.
 Select SortA in the Edit menu and enter.
 The calculator will ask for the list. Type in the list
(L1), close the parentheses, and enter.
 The calculator will say Done.
 Press stat, select edit in the Edit menu, and enter.
 You will see that your data in L1 has been sorted in an
increasing order.
 If the data size is odd, the median is the middle value.
If the data size is even, the median is the average of the
middle two values.

Computing the mean
if only raw data is given
 Enter your data in a list, say L1.
 Select "1Var Stats" in the CALC menu and enter.
 The calculator will ask for the list. Type in the list
L1 and enter.
 The calculator will give a list of numbers; xbar is the
mean x.

Computing the mean
if the frequency table is given
 Enter the frequency table in the calculator, say, xvalues
in L1 and frequencies in L2.
 Select "1Var Stats" in the CALC menu and enter.
 The calculator will ask for the lists. Type in the list
L1, L2 and enter.
 The calculator will give a list of numbers; xbar is the
mean x.

Computing the
meadian
Do the same as above and scroll down.

Problems on 2.2: Mean
and Median
Exercise 2.1.1. 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: Use TI84.
Exercise 2.1.2. The following figures refer to the GPA of six
students.
Find the median and mean GPA.
Exercise 2.1.3. 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: Use TI84.
Exercise 2.1.4. 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: Use TI84.
Exercise 2.1.5. 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: Use TI84.
Exercise 2.1.6. 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: Use TI84.
Exercise 2.1.7. 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: Use TI84.
Exercise 2.1.8. 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: Use TI84.
2.2 Measures of Dispersion
The measures of central tendencies—mean, median, mode—
represent the middle values of the data set.
The variability of data would also be of our interest,
for a better understanding of the distribution of data.
Two data sets may have same mean and median, but they
may be spread out differently. Following is an example.
Example 2.2.1. 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.
Medians of both the sets are same —82.
But the data sets are differently dispersed.
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.
 Note that we denote the sample variance as the square of a number
s.
 Also note that we divide by n1, not by n. For some statistical
reason, dividing
by n1 works better.
 We would like our measure of dispersion to have the same units as
our data, but our formula involves squares (x_{i}x)^{2}. Therefore,
the unit of the variance, s^{2}, is the unit of the
data squared. If the data is in feet, the variance is in square feet.
To solve this problem we define another measure of dispersion, standard
deviation denoted s.
Definition. The sample
standard deviation
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)
We do some computations with the above example
2.2.1.
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 clearly higher,
the mean deviation is also so.
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
Under normal circumstances, which we will discuss later,
the mean and the standard deviation carries an enormous amount of information
regarding the distribution of the data.
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:
 At least 0 percent of the observations will fall within 1 standard
deviation of the mean, i.e, within (xs, x+s).
This is clearly obvious.
 At least 75 percent of the observations will fall within 2 standard
deviations of the mean, i.e., within (x2s,
x+2s).
 At least 89 percent of the observations will fall within 3 standard
deviations of the mean, i.e., within (x3s,
x+3s).
 More generally, at least 100(1  1/k^{2}) percent of the data
will be within k standard deviations from the mean, i.e. within (xks,
x+ks).
Chebyshev's Rule makes no assumption about the data or the variable.
If we make some reasonable
assumptions about the data, then we can improve substantially
above rule as follows.
The Empirical Rule: Suppose the histogram or the bargraph
of the data is symmetric around the vertical line through the mean x = x
as follows:
In other words, the histogram or the bargraph
should fit into a bellshaped curve like the following.
Then we have the following:
 Approximately 68.3 percent of the observations will fall in the interval
(xs, x+s).
 Approximately 95.4 percent of the observations will fall in the interval
(x2s, x+2s).
 Approximately 99.7 percent of the observations will fall within the
interval (x3s, x+3s).
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 is given in a frequency table (ungrouped). Let x_{i}
denote the values and f_{i} be the frequency of x_{i}.
Then
 the mean =
 the variance =
 A simplified formula for variance is
s^{2}
= 
1
n 1

[∑

(f_{i}x_{i}^{2}) 
n x^{2}

]. 
 If the data is given in a class
frequency table of the grouped
data, we use the same formula, with x_{i}
as the class mark, which is the average of the class limits.
In this case, we only get an estimate of the variance of the original data.
Example 2.2.2. 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.
Remark: When computing power was not
in abundance
(only 20 or 30 years ago), as it is now, we used to compute mean and variance using such tables to
do the computations.
Such methods are out of date by now. We use TI84 or other tools now.
Example 2.2.3. Following is the class frequency
distribution of the data on birth weight of some babies (exercise 1.1,
Chapter 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.
 Note that we can only get an approximate mean and variance if we
use the class mark and with the above formula. If you use the
original data, you will notice a difference.
 As mentioned above,
because of the availability of computers, the importance of such
approximations has declined.
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 (TI84): 
Computing the variance
and standard deviation
 Follow the same steps used for computing the mean (using
either raw data or the frequency table).
 The calculator will give a list of numbers; S_{X}
is the standard deviation.
 The variance is the square of the standard deviation.

Problems on 2.2: Variance,
Standard Deviation, and Use of the Frequency Table
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 variance and standard deviation of the price.
Solution: Use TI84.
Exercise 2.2.2. The following figures refer
to the GPA of six students.
Find the variance and standard deviation of GPA.
Exercise 2.2.3. 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: Use TI84.
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 variance and standard deviation of time taken by the athlete.
Solution: Use TI84.
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 variance and standard deviation of the weight, at birth,
of these babies.
Solution: Use TI84.
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 variance and standard deviation of the hourly wages.
Solution: Use TI84.
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 number, variance, and standard deviation of typos in
a book.
Solution: Use TI84.
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 variance and standard deviation of the length, at birth,
of these babies.
Solution: Use TI84.
Exercise 2.2.9. 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: Use TI84.
Exercise 2.2.10. 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: Use TI84.
