A survey regarding the heights (in cm) of 50 girls of class X of a school was conducted and the following data was obtained :
| Height (in cm) | Number of girls |
|---|---|
| 120 - 130 | 2 |
| 130 - 140 | 8 |
| 140 - 150 | 12 |
| 150 - 160 | 20 |
| 160 - 170 | 8 |
| Total | 50 |
Find the mean and mode of the above data.
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Step-by-Step Solution
Step 1: Calculate Class Marks and Mean
To calculate the mean of grouped data, we first find the class mark (xi) for each class interval by taking the midpoint of the interval. Then, we multiply each class mark by its corresponding frequency (fi) to get fixi. Finally, we sum all fixi values and divide by the total frequency (∑fi).
Step 2: Calculate the Mean
Substitute the calculated sum of fixi and the total frequency into the mean formula to find the mean height.
Step 3: Identify Modal Class and Parameters
The modal class is the class interval with the highest frequency. From the given data, the class 150−160 has the highest frequency of 20. We then identify the lower limit (l), class size (h), frequency of the modal class (f1), frequency of the class preceding the modal class (f0), and frequency of the class succeeding the modal class (f2).
Step 4: Apply Mode Formula
The mode for grouped data is calculated using the formula: l+(2f1−f0−f2f1−f0)×h. This formula helps to estimate the mode within the modal class.
Step 5: Calculate the Mode
Substitute the identified values into the mode formula and perform the calculations to find the mode.