Arithmetic Mean for Frequency Distribution – Statistics

Arithmetic Mean
The arithmetic mean is the amount secured by dividing the sum of values of the items in a series by the number.
1. Arithmetic Mean for Ungrouped Data
If n numbers, x₁,x₂,…,x_n, then their arithmetic mean or their average

2. Arithmetic Mean for Frequency Distribution
Let f₁,f₂,…,f_n be corresponding frequencies of x₁,x₂,…,x_n. Then,

3. Combined Mean
The number of value A₁ is n₁. The number of value A₂ is n₂. … . The number of value A_r is n_r. Then the combined mean is given by

It looks like the Arithmetic Mean for Frequency Distribution.

Mean of Grouped Data
The mean (or average) of observations, as we know, is the sum of the values of all the observations divided by the total number of observations. From Class IX, recall that if x₁, x₂, …, x_n, are observations with respective frequencies f₁, f₂, …, f_n, then this means observation x₁ occurs f₁ times, x₂ occurs f₂ times, and so on.
Now, the sum of the values of all the observations =f₁ x₁+f₂ x₂+⋯+f_n x_n, and
the number of observations =f₁+f₂+⋯+f_n.
So, the mean x̄ of the data is given by

Recall that we can write this in short form by using the Greek letter I (capital
sigma) which means summation. That is,

which, more briefly, is written as

, if it is understood that i varies from 1 to n.
Let us apply this formula to find the mean in the following example.

Examples: Questions and Solutions
Q1. Calculate the mean for the following distribution:

Q2. The ages of 40 students are given in the following table:

Find the arithmetic mean.
solution:

Q3. Find the mean of the following data:

Q4. The table below represents Mathematics test scores and frequency for each score.

(a) Determine the median
(b) Determine the mean
solution:
(a) Σf=25
i.e. there are 25 scores. To determine the median, find the position of the median by adding the frequencies until you reach the position of the median.
Median lies in position 13, hence median =20
(b) mean

Q5. From the data given below, calculate the mean wage, correct to the nearest rupee.

(i) If the number of workers in each category is doubled, what would be the new mean wage?
(ii) If the wages per day in each category are increased by 60%; what is the new mean wage?
(iii) If the number of workers in each category is doubled and the wages per day per worker are reduced, what is the new mean wage?
solution:

(i) Mean remains the same if the number of workers in each category is doubled.
Mean =80
(ii) Mean will be increased by 60% if the wages per day per worker is increased by 60%

(iii) No change in the mean if the number of workers is doubled but if wages per worker is reduced by 40% then

Q6. The marks obtained by 30 students of Class X of a certain school in a Mathematics paper consisting of 100 marks are presented in table below. Find the
mean of the marks obtained by the students.

solution:
Recall that to find the mean marks, we require the product of each x_i with the corresponding frequency f. So, let us put them in a column as shown in this table.