Definition of standard deviation from the mean:
In Mathematics or Statistics, the standard deviation is a measure of dispersion from the mean. It tells us as to how far is the data spread from the mean value.
Calculating standard deviation of the mean:
The following steps are to be followed to calculate the standard deviation for a set of data {x1, x2, x3, x4….Xn} with mean = Xm.
Step 1: Make a table as follows:
Step 2: Add a column to the table for Xi – Xm
Step 3: Next square the difference thus found.
Step 4: Find the sum of the values of the third column
= ∑(Xi – Xm)^2
Step 5: Divide the value obtained in step 4 by number of observations n
= [∑(Xi – Xm)^2]/n
Step 6: Standard deviation = SD = square root of the value found in step 5
Therefore,
SD = √{[∑(Xi – Xm)^2]/n}
Significance of the standard deviation from mean:
A measure of dispersion is a number showing how scattered the observations are from the mean. We already established above that standard deviation is a measure of dispersion.
If the difference between the observations and their mean are small, that is, if the observations are not too far away scattered from the mean, then the value of the standard deviation number is small. Such a data can be regarded as consistent or stable. In such cases, the mean can be considered as a good representative of the observations.
However, if the differences between the observations and their mean are large, that is, if the observations scattered far away from the mean, then the value of standard deviation would be large. Such a data cannot be regarded as stable and in some cases it may also be inconsistent. For such a data, the mean cannot be regarded as a good representative of the observations.
For example, consider the marks of two students X and Y in five subjects, as follows:
The mean marks of both the students are the same but since student X has a lower standard deviation, we can say that he is a more consistent performer.
In Mathematics or Statistics, the standard deviation is a measure of dispersion from the mean. It tells us as to how far is the data spread from the mean value.
Calculating standard deviation of the mean:
The following steps are to be followed to calculate the standard deviation for a set of data {x1, x2, x3, x4….Xn} with mean = Xm.
Step 1: Make a table as follows:
Xi
|
X1
|
X2
|
X3
|
:
|
:
|
Xn
|
Step 2: Add a column to the table for Xi – Xm
Xi
|
Xi – Xm
|
X1
|
X1 – Xm
|
X2
|
X2 – Xm
|
X3
|
X3 – Xm
|
:
|
|
:
|
|
:
|
|
Xn
|
Xn – Xm
|
Step 3: Next square the difference thus found.
Xi
|
Xi – Xm
|
(Xi – Xm)^2
|
X1
|
X1 – Xm
|
(X1 – Xm)^2
|
X2
|
X2 – Xm
|
(X2 – Xm)^2
|
X3
|
X3 – Xm
|
(X3 – Xm)^2
|
:
|
:
|
:
|
:
|
:
|
:
|
:
|
:
|
:
|
Xn
|
Xn - Xm
|
(Xn – Xm)^2
|
Step 4: Find the sum of the values of the third column
= ∑(Xi – Xm)^2
Step 5: Divide the value obtained in step 4 by number of observations n
= [∑(Xi – Xm)^2]/n
Step 6: Standard deviation = SD = square root of the value found in step 5
Therefore,
SD = √{[∑(Xi – Xm)^2]/n}
Significance of the standard deviation from mean:
A measure of dispersion is a number showing how scattered the observations are from the mean. We already established above that standard deviation is a measure of dispersion.
If the difference between the observations and their mean are small, that is, if the observations are not too far away scattered from the mean, then the value of the standard deviation number is small. Such a data can be regarded as consistent or stable. In such cases, the mean can be considered as a good representative of the observations.
However, if the differences between the observations and their mean are large, that is, if the observations scattered far away from the mean, then the value of standard deviation would be large. Such a data cannot be regarded as stable and in some cases it may also be inconsistent. For such a data, the mean cannot be regarded as a good representative of the observations.
For example, consider the marks of two students X and Y in five subjects, as follows:
Subject
|
Student X
|
Student Y
|
A
|
58
|
86
|
B
|
52
|
39
|
C
|
45
|
12
|
D
|
48
|
68
|
E
|
57
|
55
|
Mean
|
52
|
52
|
SD
|
5.612
|
28.24
|
The mean marks of both the students are the same but since student X has a lower standard deviation, we can say that he is a more consistent performer.
