Standard Deviation

Standard Deviation

How to find the standard deviation of a data: formula, examples, and their solutions.

Formula

The standard deviation is the square root of the variance. So, just like the variance, the standard deviation also means how far the values are from the mean.

The standard deviation σ(X)
is the square root of the variance V(X).

So, just like the variance,
the standard deviation also means
how far the values are from the mean x.

The reason to write a square root
is to undo the effect of the square in V(X):
(xi - x)2.

By adding a square root,
the unit of σ(X)
and the unit of x
are the same.

So it's convineient to compare
the standard deviation and the mean.

Example 1: Standard Deviation of 30, 40, 50, 60, 70

The following data show 5 test scores of a student. 30, 40, 50, 60, 70. Find the standard deviation of the data.

First, find the mean of the data.

The values are 30, 40, 50, 60, 100.
There are 5 values.

So the mean x is
(30 + 40 + 50 + 60 + 70)/5.

Mean (Average)

30 + 70 = 100
40 + 60 = 50

100 + 100 + 50 = 250

250/5 = 50

So x = 50.

Draw a three column table like this.

Write the titles
xi, xi - x, and (xi - x)2.

Write the given values in the xi column.

x = 50

So write the value of xi - x, xi - 50,
in the next column.

30 - 50 = -20
40 - 50 = -10
50 - 50 = 0
60 - 50 = 10
70 - 50 = 20

Write the squares of the second column
in the (xi - x)2 column.

(-20)2 = 400
(-10)2 = 100
02 = 0
102 = 100
202 = 400

Then write the sum of the third column
in the next row:
400 + 100 + 0 + 100 + 400 = 1000.

This 1000 is the sum of (xi - x)2.

The sum of (xi - x)2 is 1000.
n = 5

So V(X) = 1000/5.

1000/5 = 200

So V(X) = 200.

The standard deviation σ(X)
is the square root of the variance V(X).

V(X) = 200

So σ(X) = √200.

200 = 2⋅100

100 = 102

Take 10 out from the square root sign.

Then σ(X) = 10√2.

Simplify a radical

x = 50 [points]
V(X) = 200 [points2]
σ(X) = 10√2 [points]

Unlike the unit of V(X) [points2],
the unit of σ(X) [points]
is the same as the unit of x [points].

This is why
the standard deviation is widely used
in analyzing data.

Example 2: Standard Deviation from a Frequency Table

The following table shows quiz scores of 20 students. Find the standard deviation of the data.

If a frequency table is given,
the solution is quite different.

First, make a six column table.

Write the titles
xi (score), fi (frequency),
xifi, xi - x, (xi - x)2, and fi(xi - x)2.

Write the scores in the xi column:
0, 1, 2, 3, 4, 5.

And write the frequencies in the fi column:
1, 1, 4, 7, 5, 2.

Then write the sum of the fi column
in the ∑ row:
1 + 1 + 4 + 7 + 5 + 2 = 20.

So n = 20.

Write xifi column
by multiplying xi and fi.

0⋅1 = 0
1⋅1 = 1
2⋅4 = 8
3⋅7 = 21
4⋅5 = 20
5⋅2 = 10

Write the sum of the xifi column
in the ∑ row:
0 + 1 + 8 + 21 + 20 + 10 = 60.

This 60 is the sum of the values.

(sum) = 60
n = 20

So x = 60/20.

So x = 3.

Mean from the Frequency Table

x = 3

So write the value of xi - x, xi - 3,
in the next column.

0 - 3 = -3
1 - 3 = -2
2 - 3 = -1
3 - 3 = 0
4 - 3 = 1
5 - 3 = 2

Write (xi - x)2 column
by squaring the (xi - x) column.

(-3)2 = 9
(-2)2 = 4
(-1)2 = 1
02 = 0
12 = 1
22 = 4

Write fi(xi - x)2 column
by multiplying fi and (xi - x)2.

1⋅9 = 9
1⋅4 = 4
4⋅1 = 4
7⋅0 = 0
5⋅1 = 5
2⋅4 = 8

Write the sum of the fi(xi - x)2 column
in the ∑ row:
9 + 4 + 4 + 0 + 5 + 8 = 30.

This 30 is the total sum of (xi - x)2.

The total sum of (xi - x)2 is 30.

n = 20

So V(X) = 30/20.

30/20 = 3/2

So V(X) = 3/2.

σ(X) = √V(X)

V(X) = 3/2

So σ(X) = √3/2.

3/2 = √3/√2

Divide radicals

To rationalize the denominator √2,
multiply √2/√2.

Then σ(X) = √6/2.

Rationalizing a Denominator