**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, *x*_{1},*x*_{2},…,*x*_{n}, then their arithmetic mean or their average

2.

**Arithmetic Mean for Frequency Distribution**

Let

*f*

_{1},

*f*

_{2},…,

*f*

_{n}be corresponding frequencies of

*x*

_{1},

*x*

_{2},…,

*x*

_{n}. Then,

3. Combined Mean

The number of value

*A*

_{1}is

*n*

_{1}. The number of value

*A*

_{2}is

*n*

_{2}. … . 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 *x*_{1}, *x*_{2}, …, *x*_{n}, are observations with respective frequencies *f*_{1}, *f*_{2}, …, *f*_{n}, then this means observation *x*_{1} occurs *f*_{1} times, *x*_{2} occurs *f*_{2} times, and so on.

Now, the sum of the values of all the observations =*f*_{1} *x*_{1}+*f*_{2} *x*_{2}+⋯+*f*_{n} *x*_{n}, and

the number of observations =*f*_{1}+*f*_{2}+⋯+*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) Σ

*f*=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.

(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.

Q7. If the mean of the following data is 15, find

*p*.

Q8. Find the value of

*p*for the following distribution whose mean is 16.6.

Q9. The following table gives the heights of plants in centimeter. If the mean height of plants is 60.95 cm: find the value of ‘

*f*‘.

Q10. If the mean of the following data is 20.6. Find the value of

*p*.

Q11. The marks obtained by 40 students in a short assessment is given below, where

*a*and

*b*are two missing data.

If mean of the distribution is 7.2. find

*a*and

*b*.

solution:

Mean

Total number of students

Let’s read the post ‘The Mean of Continuous Data or Discrete Data (Grouped Data)’.