Question

Find the arithmetic mean from the following table:

Hint:

To find the arithmetic mean from a frequency distribution, use the formula \( \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \), where \( f_i \) is the frequency and \( x_i \) is the midpoint of each class interval.

Answer 1

To find the arithmetic mean of the frequency distribution, we use the formula: \[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i}, \] where \( f_i \) is the frequency and \( x_i \) is the class mark (midpoint). Step 1: Calculate the cumulative frequency
We are given the cumulative frequencies directly, so we can calculate the frequency for each class:
- Number of students who obtained less than 5: 3,
- Number of students who obtained less than 10: 10,
- Number of students who obtained less than 15: 25,
- Number of students who obtained less than 20: 49,
- Number of students who obtained less than 25: 65,
- Number of students who obtained less than 30: 73,
- Number of students who obtained less than 35: 78,
- Number of students who obtained less than 40: 80.
Thus, the frequency for each class is:
- \( f_1 = 3 \) (for less than 5),
- \( f_2 = 7 \) (10 - 3 for less than 10),
- \( f_3 = 15 \) (25 - 10 for less than 15),
- \( f_4 = 24 \) (49 - 25 for less than 20),
- \( f_5 = 16 \) (65 - 49 for less than 25),
- \( f_6 = 8 \) (73 - 65 for less than 30),
- \( f_7 = 5 \) (78 - 73 for less than 35),
- \( f_8 = 2 \) (80 - 78 for less than 40).
Step 2: Calculate the midpoints for each class interval
The class intervals are:
- Less than 5: midpoint \( x_1 = 2.5 \),
- Less than 10: midpoint \( x_2 = 7.5 \),
- Less than 15: midpoint \( x_3 = 12.5 \),
- Less than 20: midpoint \( x_4 = 17.5 \),
- Less than 25: midpoint \( x_5 = 22.5 \),
- Less than 30: midpoint \( x_6 = 27.5 \),
- Less than 35: midpoint \( x_7 = 32.5 \),
- Less than 40: midpoint \( x_8 = 37.5 \).
Step 3: Calculate \( \sum f_i x_i \)
Now, calculate \( f_i x_i \) for each class:
- \( f_1 x_1 = 3 \times 2.5 = 7.5 \),
- \( f_2 x_2 = 7 \times 7.5 = 52.5 \),
- \( f_3 x_3 = 15 \times 12.5 = 187.5 \),
- \( f_4 x_4 = 24 \times 17.5 = 420 \),
- \( f_5 x_5 = 16 \times 22.5 = 360 \),
- \( f_6 x_6 = 8 \times 27.5 = 220 \),
- \( f_7 x_7 = 5 \times 32.5 = 162.5 \),
- \( f_8 x_8 = 2 \times 37.5 = 75 \).
Now, sum up \( f_i x_i \): \[ \sum f_i x_i = 7.5 + 52.5 + 187.5 + 420 + 360 + 220 + 162.5 + 75 = 1535. \] Step 4: Calculate the total frequency The total frequency is: \[ \sum f_i = 3 + 7 + 15 + 24 + 16 + 8 + 5 + 2 = 80. \] Step 5: Calculate the arithmetic mean
Finally, substitute the values into the formula for the arithmetic mean: \[ \bar{x} = \frac{1535}{80} = 19.1875. \]
Conclusion: The arithmetic mean is \( 19.19 \) (rounded to two decimal places).