Frequency Distribution Practice Questions

Frequency Distribution is a method of organizing data that shows how often each value or group of values occurs in a dataset.

Example 1: Suppose we have a series with a mean of 20 and a variance of 100. Find out the Coefficient of Variation. 

Solution: 

We know the formula for Coefficient of Variation, 

\frac{\sigma}{\bar{x}} \times 100

Given mean \bar{x} = 20 and variance \sigma^2 = 100. 

We know ,

Standard Deviation \sigma=\sqrt{varience}=\sqrt{100}

Standard Deviation \sigma=10

Substituting the values in the formula,

\frac{\sigma}{\bar{x}} \times 100 \\ = \frac{10}{20} \times 100 \\ = \frac{10}{20} \times 100 \\ = 50

Example 2: Given two series with Coefficients of Variation of 70 and 80. The means are 20 and 30. Find the values of the standard deviation for both series.

Solution: 

In this question we need to apply the formula for CV and substitute the given values. 

Standard Deviation of first series. 

C.V = \frac{\sigma}{\bar{x}} \times 100 \\ 70 = \frac{\sigma}{20} \times 100 \\ 1400 = \sigma \times 100 \\ \sigma=14

Thus, the standard deviation of first series = 14

Standard Deviation of second series. 

C.V = \frac{\sigma}{\bar{x}} \times 100 \\ 80 = \frac{\sigma}{30} \times 100 \\ 2400 = \sigma \times 100 \\ \sigma=24

Thus, the standard deviation of first series = 24

Example 3: Draw the frequency distribution table for the following data: 2, 3, 1, 4, 2, 2, 3, 1, 4, 4, 4, 2, 2, 2

Solution: 

Since there are only very few distinct values in the series, we will plot the ungrouped frequency distribution. 

Value	Frequency
1	2
2	6
3	2
4	4
Total	14

Example 4: The table below gives the values of temperature recorded in Hyderabad for 25 days in summer. Represent the data in the form of a less-than-type cumulative frequency distribution: 

Solution: 

Since there are so many distinct values here, we will use grouped frequency distribution. Let's say the intervals are 20-25, 25-30, 30-35. Frequency distribution table can be made by counting the number of values lying in these intervals. 

Temperature	Number of Days
20-25	2
25-30	10
30-35	13

This is the grouped frequency distribution table. It can be converted into cumulative frequency distribution by adding the previous values. 

Temperature	Number of Days
Less than 25	2
Less than 30	12
Less than 35	25

Example 5: Make a Frequency Distribution Table for the data:

{45, 22, 37, 18, 56, 33, 42, 29, 51, 27, 39, 14, 61, 19, 44, 25, 58, 36, 48, 30, 53, 41, 28, 35, 47, 21, 32, 49, 16, 52, 26, 38, 57, 31, 59, 20, 43, 24, 55, 17, 50, 23, 34, 60, 46, 13, 40, 54, 15, 62}

Solution:

To create the frequency distribution table for given data, let's arrange the data in ascending order as follows:

{13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62}

Now, we can count the observations for intervals: 10-20, 20-30, 30-40, 40-50, 50-60 and 60-70.

Interval	Frequency
10 - 20	7
20 - 30	10
30 - 40	10
40 - 50	10
50 - 60	10
60 - 70	3