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?
Number of wicketsNumber of bowlers20-60760-1005100-15016150-25012250-3502350-4503
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Step-by-Step Solution
Step 1: Calculate Class Marks (xi)
To find the mean, we first need to determine the class mark (midpoint) for each class interval. The class mark xi 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 (xi) 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 xi 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 (di)
Next, we calculate the deviation di for each class mark. This is done by subtracting the assumed mean (a) from each class mark (xi). These deviations indicate how far each class mark is from our assumed average.
Step 4: Calculate Step Deviations (ui)
To simplify calculations, especially when deviations are large, we divide each deviation di by the chosen class size (or common divisor) h to get the step deviation ui. This makes the numbers smaller and easier to work with.
Step 5: Calculate fiui and Sum Frequencies
We multiply the frequency (fi) of each class by its corresponding step deviation (ui) to get fiui. We also sum all the frequencies (∑fi) to get the total number of bowlers.
Step 6: Calculate Sum of fiui
Now we sum all the fiui values calculated in the previous step. This sum (∑fiui) 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 fiui, the sum of frequencies ∑fi, and the chosen divisor h into the formula. This method is efficient for grouped data with large xi values and varying class sizes.
Step 8: Calculate the Mean
Substitute the calculated values into the formula: a=200, ∑fiui=−44, ∑fi=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.