Standard Deviation Frequency Table

Standard deviation is an important study in Statistics. Standard deviation is the square root of the mean of the squared deviations divided by number of data. The notation of Standard deviation is σ.

The formula for calculating standard deviation is

Standard deviation(σ) =` sqrt((sum_(i=1)^n (fx^2))-bar(x)^2)/n `

Here, n = `sum` f  number of data

In this article, we discuss about calculating standard deviation from frequency table.

Steps to Calculate Standard Deviation from Frequency Table:

Step 1: In frequency table, we calculate sum of fx2.

Step 2: Then we calculate mean bar(x) from frequency table.

Mean `bar(x)` = `(sum fx)/"n"`

Step 3: Calculate standard deviation using formula.

Let us see example problems for calculating standard deviation.

Example 1:

Calculate the standard deviation from the frequency table.

X = Weight(kg)	40	50	60	80
F = Frequency	6	8	10	16

Solution: 

The frequency table is,

Weight (x)	Frequency (f)	fx	x2	fx2
40	6	240	1600	9600
50	8	400	2500	20000
60	10	600	3600	36000
80	16	1280	6400	102400
Total	40	2520		168000

Now we are going to calculate mean bar(x) using  formula,

Mean `bar(x)` = `(sum fx)/n`

From frequency table, we know that,

`bar(x)` = `2520/40`

`bar(x)` = 63

Now we are going to calculate standard deviation from formula.

Standard deviation(σ) = `sqrt((sum_(i=1)^n (fx^2))-bar(x)^2)/n `

From frequency table, we know that,

Standard deviation(σ) = `sqrt((168000/40)-63^2)`

Standard deviation =  `sqrt(4200-3969)`

Standard deviation = `sqrt(231)`

Standard deviation = 15.2

Therefore, standard deviation of frequency table is 15.2.

Another Example Problem for Calculating Standard Deviation from Frequency Table:

Example 2:

Calculate the standard deviation from the frequency table.

X = Marks	30	40	50	60
F = Frequency	5	7	9	15

Solution: 

The frequency table is,

 

Marks (x)	Frequency (f)	fx	x2	fx2
10	10	100	100	1000
20	11	220	400	4400
30	14	420	900	12600
40	16	640	1600	25600
Total	51	1380		43600

Now we are going to calculate mean bar(x) using  formula,

Mean `bar(x)` = `(sum fx)/n`

From frequency table, we know that,

`bar(x)` = `1380/51`

`bar(x)` = 27.1

Now we are going to calculate standard deviation from formula.

Standard deviation(σ) = `sqrt((sum_(i=1)^n (fx^2))-bar(x)^2)/n `

From frequency table, we know that,

Standard deviation(σ) =` sqrt((43600/51)-(27.1)^2)`

Standard deviation =  `sqrt(854.9-734.4)`

Standard deviation = `sqrt(120.5)`

Standard deviation = 10.9

Therefore, standard deviation of frequency table is 10.9.