The distribution below shows the number of wickets taken by bowlers in one-day cricket matches. Find the mean number of wickets by choosing a suitable method. What does the mean signify? $\begin{array}{|l|c|c|c|c|c|c|}\hline \text{Number of wickets} & 20\text{-}60 & 60\text{-}100 & 100\text{-}150 & 150\text{-}250 & 250\text{-}350 & 350\text{-}450 \\ \hline \text{Number of bowlers} & 7 & 5 & 16 & 12 & 2 & 3 \\ \hline \end{array}$

Step 1: Calculate Class Marks ($x_i$)

To find the mean, we first need to determine the class mark (midpoint) for each class interval. The class mark $x_i$ is calculated as the average of the lower and upper limits of each class interval. We will use these class marks in further calculations to represent the values within each interval.

Step 2: Choose Assumed Mean ($a$) and Class Size ($h$)

Since the class sizes are unequal and the midpoints ($x_i$) are large, the step-deviation method is suitable as suggested by the textbook. We choose an assumed mean $a = 200$ which is roughly in the middle of our $x_i$ values. For $h$, we can choose a suitable divisor for the deviations, even if class sizes vary, as demonstrated in Example 3 of the textbook. Here, $h=20$ is a suitable common divisor.

Step 3: Calculate Deviations ($d_i$)

Next, we calculate the deviation $d_i$ for each class mark. This is done by subtracting the assumed mean ($a$) from each class mark ($x_i$). These deviations indicate how far each class mark is from our assumed average.

Step 4: Calculate Step Deviations ($u_i$)

To simplify calculations, especially when deviations are large, we divide each deviation $d_i$ by the chosen class size (or common divisor) $h$ to get the step deviation $u_i$. This makes the numbers smaller and easier to work with.

Step 5: Calculate $f_i u_i$ and Sum Frequencies

We multiply the frequency ($f_i$) of each class by its corresponding step deviation ($u_i$) to get $f_i u_i$. We also sum all the frequencies ($\sum f_i$) to get the total number of bowlers.

Step 6: Calculate Sum of $f_i u_i$

Now we sum all the $f_i u_i$ values calculated in the previous step. This sum ($\sum f_i u_i$) is a crucial component for applying the step-deviation formula for the mean.

Step 7: Apply Step-Deviation Formula for Mean

We use the step-deviation formula to find the mean. We substitute the values of the assumed mean $a$, the sum of $f_i u_i$, the sum of frequencies $\sum f_i$, and the chosen divisor $h$ into the formula. This method is efficient for grouped data with large $x_i$ values and varying class sizes.

Step 8: Calculate the Mean

Substitute the calculated values into the formula: $a = 200$, $\sum f_i u_i = -44$, $\sum f_i = 45$, and $h = 20$. Perform the arithmetic operations to find the numerical value of the mean. The result is approximately 180.44.

Step 9: Significance of the Mean

The calculated mean of approximately 180.44 signifies that, on average, the bowlers in this distribution have taken about 180.44 wickets in one-day cricket matches. It provides a single representative value for the entire data set.

More from this chapter

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