1. CHAPTER 2: Basic Summary Statistics
Measures of Central Tendency (or location)
Mean – mode – median
Measures of Dispersion (or Variation)
Variance – standard deviation – coefficient of variation
2.1. Introduction:
For the population of interest, there is a population of values of
the variable of interest.
2. (i) A parameter is a measure (or number) obtained from
the population values X1,X2, …, XN
(parameters are unknown in general)
A statistic is a measure (or number) obtained from the sample
values x1,x2, …, xn
(statistics are known in general)
Let X1,X2, …, XN be the population values (in general, they are
unknown) of the variable of interest. The population size = N
Let x1,x2, …, xn be the sample values (these values are known)
The sample size = n
3. 2.2. Measures of Central Tendency: (Location)
The values of a variable often tend to be concentrated around
the center of the data.
Some of these measures are: the mean, mode, median
These measures are considered as representatives (or typical values)
of data.
Mean:
:
(1) Population mean :
If X1,X2, …, XN are the population values of the variable of
interest , then the population mean is:
N
N
i
i
X
N
N
X
X
X
1
2
1
(unit)
4. The population mean is a parameter (it is usually unknown)
(2) Sample mean :
If are the sample values, then the sample mean is
x
n
x
x
x ,
,
, 2
1
n
x
n
x
x
x
x
n
i
i
n
1
2
1
(unit)
5. The sample mean is a statistic (it is known)
The sample mean is used to approximate (estimate) the population
mean .
x
x
Example:
Consider the following population values:
Suppose that the sample values obtained are:
.
41
,
27
,
35
,
22
,
30 5
4
3
2
1
X
X
X
X
X
.
27
,
35
,
30 3
2
1
x
x
x
6. Notes:
The mean is simple to calculate.
There is only one mean for a given sample data.
The mean can be distorted by extreme values.
The mean can only be found for quantitative variables
Median:
The median of a finite set of numbers is that value which
divides the ordered set into two equal parts.
Then:
31
5
155
5
41
27
35
22
30
67
.
30
3
92
3
27
35
30
x
(unit)
(unit)
Let x1,x2, …, xn be the sample values . We have two cases:
7. (1) If the sample size, n, is odd:
The median is the middle value of the ordered
observations.
* *
1 2
…
…
Middle
value=
MEDIAN
2
1
n …
… *
n
Ordered set
(smallest to
largest)
Rank (or order)
2
1
n
The middle observation is the ordered observation
The median = The order observation.
2
1
n
8. Example:
Find the median for the sample values: 10, 54, 21, 38, 53.
Solution:
n = 5 (odd number)
The rank of the middle value (median) = = (5+1)/2 = 3
2
1
n
10 21 38 53 54
2
1
n
1 2 = 3 4 5
Ordered set
Rank (or order)
The median =38 (unit)
9. (2) If the sample size, n, is even:
The median is the mean (average) of the two middle values of the
ordered observations.
The middle two values are the ordered and observations.
2
n
1
2
n
The median =
Ordered set
Rank (or order)
10. Example:
Find the median for the sample values: 10, 35, 41, 16, 20, 32
Solution:
.n = 6 (even number)
The rank of the middle values are
= 6 / 2 = 3
2
n
= (6 / 2) + 1 = 4
1
2
n
11. Ordered set
Rank (or order)
The median (unit)
26
2
52
2
32
20
Note:
The median is simple to calculate.
There is only one median for given data.
he median is not affected too much by extreme values.
The median can only be found for quantitative variables
12. Mode:
The mode of a set of values is that value which occurs with the
highest frequency.
If all values are different or have the same frequency, there is no
mode.
A set of data may have more than one mode.
Example:
Note:
The mode is simple to calculate but it is not “good”.
The mode is not affected too much by extreme values.
The mode may be found for both quantitative and qualitative
variables.
13. 2.3. Measures of Dispersion (Variation):
The variation or dispersion in a set of values refers to how
spread out the values are from each other.
•
The variation is small when the values are close
together.
• There is no variation if the values are the same.
14. Some measures of dispersion:
Range – Variance – Standard deviation
Coefficient of variation
Range:
Range is the difference between the largest (Max) and smallest
(Min) values.
Range = Max Min
Smaller variation
Larger variation
15. Example:
Find the range for the sample values: 26, 25, 35, 27, 29, 29.
Solution:
Range = 35 25 = 10 (unit)
Note:
The range is not useful as a measure of the variation since it only
takes into account two of the values. (it is not good)
Variance:
The variance is a measure that uses the mean as a point of
reference.
The variance is small when all values are close to the mean. The
variance is large when all values are spread out from the mean.
deviations from the mean:
16. (1) Population variance:
Let X1,X2, …, XN be the population values.
The population variance is defined by
Deviations from the mean:
where is the population
mean
N
X
N
i
i
1
17. (2) Sample Variance:
Let be the sample values.
The sample variance is defined by:
n
x
x
x ,
,
, 2
1
where is the sample mean.
n
x
x
n
i
i
1
18. Example:
We want to compute the sample variance of the following sample
values: 10, 21, 33, 53, 54.
Solution: n=5
20. Note:
To calculate S2
we need:
n = sample size
The sum of the values
The sum of the squared values
i
x
2
i
x
21. 7
.
376
4
8
.
1506
1
5
2
.
34
5
7355
2
2
S
Standard Deviation:
The standard deviation is another measure of variation.
It is the square root of the variance.
(1) Population standard deviation is: (unit)
2
(2) Sample standard deviation is: (unit)
2
S
S
22. Coefficient of Variation (C.V.):
The variance and the standard deviation are useful as
measures of variation of the values of a single variable for a
single population (or sample).
If we want to compare the variation of two variables we
cannot use the variance or the standard deviation because:
1. The variables might have different units.
2. The variables might have different means.
23. We need a measure of the relative variation that
will not depend on either the units or on how large the
values are. This measure is the coefficient of variation
(C.V.) which is defined by:
C.V. = (free of unit or unit less)
%
100
*
x
S
1
x 1
S %
100
.
1
1
1
x
S
V
C
2
x 2
S %
100
.
2
2
2
x
S
V
C
Mean St.dev. C.V.
1st
data set
2nd
data set
24. The relative variability in the 1st
data set is larger than
the relative variability in the 2nd
data set if C.V1
> C.V2
(and vice
versa).
Example:
1st
data set: 66 kg, 4.5 kg
2nd
data set: 36 kg, 4.5 kg
Since , the relative variability in the 2nd
data set is
larger than the relative variability in the 1st
data set.
1
x
2
S
%
8
.
6
%
100
*
66
5
.
4
. 1
V
C
2
x
2
S
%
5
.
12
%
100
*
36
5
.
4
. 2
V
C
2
1 .
. V
C
V
C
25. Notes: (Some properties of , S, and S2
:
Sample values are : x1,x2, …, xn
a and b are constants
x
n
x
x
x ,
,
, 2
1
n
ax
ax
ax ,
,
, 2
1
b
x
b
x n
,
,
,
1
b
ax
b
ax n
,
,
1
x
x
a
b
x
b
x
a
S
S
a
S
S
a
2
S
2
2
S
a
2
2
S
a
Sample Data Sample
mean
Sample
st.dev
Sample
Variance
2
S
Absolute value:
0
0
a
if
a
a
if
a
a
26. Example:
Sample Sample
mean
Sample
St..dev.
Sample
Variance
1,3,5 3 2 4
(1)
(2)
(3)
-2, -6, -10
11, 13, 15
8, 4, 0
-6
13
4
4
2
4
16
4
16
Data (1) (a = 2)
(2) (b = 10)
(3) (a = 2, b = 10)
3
2
1 2
,
2
,
2 x
x
x
10
,
10
,
10 3
2
1
x
x
x
10
2
,
10
2
,
10
2 3
2
1
x
x
x
