Find the mean from the following frequency distribution :
Show Hint
For grouped data with larger numbers, the Assumed Mean Method or Step-Deviation Method can simplify the multiplication and reduce the chance of calculation errors. However, for relatively small numbers like these, the Direct Method is efficient.
Step 1: Understanding the Concept:
To find the mean of grouped data, we use the direct method. This involves finding the midpoint (class mark) of each class interval, multiplying it by the frequency, summing these products, and finally dividing by the total number of observations (total frequency). Step 2: Key Formula or Approach:
The formula for the mean (\(\bar{x}\)) of grouped data is:
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]
where \(f_i\) is the frequency of the \(i\)-th class and \(x_i\) is the class mark of the \(i\)-th class.
The class mark is calculated as \( x_i = \frac{\text{Upper class limit} + \text{Lower class limit}}{2} \). Step 3: Detailed Explanation:
We will construct a table to perform the calculations systematically.
First, find the total frequency (\(\sum f_i\)):
\[ \sum f_i = 8 + 12 + 10 + 11 + 9 = 50 \]
Next, find the sum of the products (\(\sum f_i x_i\)):
\[ \sum f_i x_i = 40 + 180 + 250 + 385 + 405 = 1260 \]
Now, calculate the mean:
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{1260}{50} = \frac{126}{5} = 25.2 \]
Step 4: Final Answer:
The mean of the frequency distribution is 25.2.
