Class 10 Maths Statistics | Mean of Grouped Data |

**Mean of Grouped Data**

In most of our real life situations, **data** is usually so **large** that to make a meaningful study it needs to be condensed as grouped data. So, we need to convert given ungrouped data into grouped data and devise some method to find its mean.

Let’s assume that we have score of 100 students, we can represent them in ungrouped data. Presenting data in this form simplifies and condenses data and enables us to observe certain important features at a glance. This is called a grouped frequency distribution table.

It is assumed that the frequency of each class interval is centered around its mid-point .

**Class mark = (Upper class limit + Lower class limit)/ 2**

3 ways to find mean of grouped Data

- Direct Mean Method
- Assumed Mean Method
- Step Deviation mean Method

The result obtained by all the three methods is the same.

**Direct Mean Method: **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.

Sum of the values of all the observations = f_{1}x_{1} + f_{2}x_{2} + . . . + f_{n}x_{n},

Number of observations = f_{1} + f_{2} + . . . + f_{n}.

**Assumed Mean Method**: The first step is to choose one among the xi’s as the assumed mean, and denote it by ‘a’. The next step is to find the difference di between a and each of the x_{i}’s, that is, the deviation of ‘a’ from each of the x_{i}’s. The third step is to find the product of d_{i} with the corresponding f_{i}, and take the sum of all the f_{i} d’s.

**Step Deviation mean Method: **The first step is to choose one among the xi’s as the assumed mean, and denote it by ‘a’. Second step is to find u_{i} = (x_{i }- a)/h , where a is the assumed mean and h is the class size. Third step is to find fi*u_{i} for all i’s & then use the formula.

The result obtained by all the three methods is the same.

So the choice of method to be used depends on the numerical values of x_{i} and f_{i}.

- If x
_{i}and f_{i}are sufficiently small, then the direct method is an appropriate choice. - If x
_{i}and f_{i}are numerically large numbers, then we can go for the assumed mean method or step-deviation method. - If the class sizes are unequal, and xi are large numerically, we can still apply the step-deviation method by taking h to be a suitable divisor of all the d
_{i}’s.

**Numerical**: A survey was conducted in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house. (Direct Mean Method)

Solution: As per direct method.

Step 1(RED) Find xi as **Class mark = (Upper class limit + Lower class limit)/ 2** , for each i.

Step 2: (GREEN) Find f_{i}x_{i } for each i.

Step 3: Find Summation of all f_{i}x_{i}

Use the formula to get mean = 162/20 = 8.1

**Numerical**: The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs.18. Find the missing frequency f. ( Assumed Mean Method)

**Solution**:

Step 1: Let assumed mean “a” be 18. (A middle value is selected; any other value can be selected here)

Step2: Find xi as **Class mark = (Upper class limit + Lower class limit)/ 2** , for each i.

Step 3: Find the difference d_{i }between a and each of the x_{i}’s, that is, the deviation of ‘a’ from each of x_{i}’s.

Step 4: Find the product of d_{i} with the corresponding f_{i}, and take the sum of all the f_{i} d’s.

Step 5: Use the formula to find Mean

Mean = 18 + (2f-40)/(44+f)

Or 18 = 18 + (2f-40)/(44+f)

Or f = 20

**Numerical**: The table below shows the daily expenditure on food of 25 households in a locality. Find the mean daily expenditure on food by step deviation method

Solution:

Step 1: Let assumed mean “a” be 225. (A middle value is selected; any other value can be selected here)

Step2: Find xi as Class mark = (Upper class limit + Lower class limit)/ 2 , for each i.

Step 3: Find the difference d_{i }between a and each of the x_{i}’s, that is, the deviation of ‘a’ from each of x_{i}’s.

Step 4: find u_{i} = (x_{i }- a)/h , where a is the assumed mean and h is the class size.

Step 5 : Find f_{i}*u_{i} for all i’s & find sum too.

Step 6: Use the formula

Mean = 225 + 50 (-7/25)

= 211 Rs

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