Substituting a = 37.5, h = 15, Σ f_i u_i = 31, and Σ f_i = 80 into the step-deviation formula gives 37.5 + (465)/(80) = 37.5 + 5.8125 = 43.3125.

Example 6. Find mean of the following data. | Class | 0-15 | 15-30 | 30-45 | 45-60 | 60-75 | 75-90 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | frequency | 12 | 15 | 11 | 20 | 16 | 6 |

Answer: 43.3125

Step-by-step solution

Step 1: Identify Class Marks and Assumed Mean

We calculate the class mark $x_i$ for each interval as the average of its limits: $7.5, 22.5, 37.5, 52.5, 67.5$ , and $82.5$ . The common class size is $h = 15$ . We choose the assumed mean as $a = 37.5$ .

Step 2: Calculate Step Deviations $u_i$ and Products $f_i u_i$

Computing $u_i$ for each class gives $-2, -1, 0, 1, 2, 3$ . Multiplying each $u_i$ by its frequency $f_i$ , we obtain the values: $12 \times (-2) = -24$ , $15 \times (-1) = -15$ , $11 \times 0 = 0$ , $20 \times 1 = 20$ , $16 \times 2 = 32$ , and $6 \times 3 = 18$ .

Step 3: Calculate the Summations $\sum f_i$ and $\sum f_i u_i$

Summing the frequencies gives $\sum f_i = 12 + 15 + 11 + 20 + 16 + 6 = 80$ . Summing the products of frequencies and step deviations yields $\sum f_i u_i = -39 + 70 = 31$ .

Step 4: Apply Step-Deviation Formula to Find Mean

Substituting $a = 37.5$ , $h = 15$ , $\sum f_i u_i = 31$ , and $\sum f_i = 80$ into the step-deviation formula gives $37.5 + \frac{465}{80} = 37.5 + 5.8125 = 43.3125$ .

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**Example 6.** Find mean of the following data. | Class | 0-15 | 15-30 | 30-45 | 45-60 | 60-75 | 75-90 | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | **frequency** | 12 | 15 | 11 | 20 | 16 | 6 |