Standard Deviation Frequency Distribution

Standard deviation is the method of finding the mean of the given data and we find the frequency distribution of the data using two formulas respectively. We use two different formulas for finding the mean for a whole and for numbers with decimals values. The variables(x) and frequency(m) are used in the standard deviation frequency distribution topic.

Basic formulas in the standard deviation frequency distribution:

1)   Mean= Sum of Given data/total number of data.

2)   σ2 = (Σ fi di2/Σ fi

 3)Sd(σ) = √(Σ (x-`barx` )/n

 

Example 1 for Standard Deviation:

 

    Find the Standard deviation and frequency distribution for the given data.

                  5, 12, 8, 4, 3, 7, 9, 22.

  Solution:

    Step 1:

          Find the mean of the given standard deviation

           Mean= `(Sum of Given data)/(no of data)` .

                      = `(5+12+8+4+3+7+9+22)/8`

                    `barx`   =8.75

Variables

Deivation of mean

         x-`barx`

(x-`barx` )2
5 -3.75 14.0625
12 3.75 14.0625
8 -0.75 0.5625
4 -4.75 22.5625
3 -5.75 33.0625
7 -1.75 3.0625
9 0.25 0.0625
22 13.25 175.5625
Average= 21.94 32.875

           Now use the third formula given above;

                   Sd(σ) = √(Σ (x-`barx` )/n

                             =√(32.785)/8

                             =0.71

 

Examples in Standard Deviation Frequency Distribution Table:

 

Example 2:

          Find the Standard deviation frequency distribution of table for the given data.

variance(xi) 6 8 5 4
frequency(fi) 5 7 3 2

Solution:

          To find the variances in the table:

variable(xi) frequency(fi) (xi)(fi) di=xi-m di2 difi2
6 5 30 6-5.5=0.5 0.25 1.25
8 7 56 8-5.5=2.5 6.25 43.75
5 3 15 5-5.5=-0.5 0.25 0.75
4 2 8 4-5.5=-1.5 2.25 4.5
  m=5.5

`109/4`

=27.25

   

`50.25/4`

=12.56

Now, use the formula from the data’s found from the table:

     Variances (σ2) = (Σ fi di2/Σ fi

                               =` (12.56/5.5)`

                               = 2.83

     Therefore, the required answer is 2.83.

Standard deviation frequency Practice Problems:

1. In the frequency table of 40 people’s weights. Find the standard deviation from the frequency table.

Weight (kg) 
50    60  70 80   
Frequency 
 6 8 10 16

 

Answer: 10.9

2. Find an estimate of the standard deviation of the following data for the marks obtained in a test by 88 students.

Marks (x) ≤ x < 10 10 ≤ x < 20   20 ≤ x < 30 30 ≤ x < 40 40 ≤ x < 50
Frequency (f) 6 16 24 25 17

 

Answer: 11.78