Frequency 04:3 59:5 10-14:9 1516:6 2024:2

Here's the solution for the given frequency distribution:

First, let's organize the data into a frequency table with midpoints (x), frequencies (f), and necessary calculations for grouped data.

Class Interval	Midpoint (x)	Frequency (f)	$fx$	Cumulative Frequency (cf)	$x^2$	$fx^2$
0-4	2	3	6	3	4	12
5-9	7	5	35	8	49	245
10-14	12	9	108	17	144	1296
15-19	17	6	102	23	289	1734
20-24	22	2	44	25	484	968
Total		$\sum f = 25$	$\sum fx = 295$			$\sum fx^2 = 4255$

Correction: The interval "15_16" is unusual. Assuming it means 15-19 to maintain consistent class widths (or 15-16 as a distinct class). Given the previous intervals (0-4, 5-9, 10-14), a class width of 5 seems intended. If 15-16 is a class, its width is 2. Let's re-evaluate the class intervals to ensure consistency.
0-4 (width 5)
5-9 (width 5)
10-14 (width 5)
15-16 (width 2) - This breaks the pattern.
20-24 (width 5)

Given the inconsistency, I will assume the user intended standard class intervals with a width of 5, and "15_16" was a typo for "15_19". If "15_16" is strictly adhered to, the calculations for median and mode become more complex due to varying class widths. For simplicity and common practice in statistics, I will assume the class 15-19 with midpoint 17.

Let's re-calculate the table with the assumption of 15-19 for the fourth class:

Class Interval	Midpoint (x)	Frequency (f)	$fx$	Cumulative Frequency (cf)	$x^2$	$fx^2$
0-4	2	3	6	3	4	12
5-9	7	5	35	8	49	245
10-14	12	9	108	17	144	1296
15-19	17	6	102	23	289	1734
20-24	22	2	22	25	484	968
Total		$\sum f = 25$	$\sum fx = 273$			$\sum fx^2 = 4255$

Self-correction: $22 \times 2 = 44$, not 22.
Let's correct the $fx$ and $\sum fx$ for the last row.
$22 \times 2 = 44$.
$\sum fx = 6 + 35 + 108 + 102 + 44 = 295$.
The $\sum fx^2$ is correct: $12 + 245 + 1296 + 1734 + 968 = 4255$.

Corrected Table:

Class Interval	Midpoint (x)	Frequency (f)	$fx$	Cumulative Frequency (cf)	$x^2$	$fx^2$
0-4	2	3	6	3	4	12
5-9	7	5	35	8	49	245
10-14	12	9	108	17	144	1296
15-19	17	6	102	23	289	1734
20-24	22	2	44	25	484	968
Total		$\sum f = 25$	$\sum fx = 295$			$\sum fx^2 = 4255$