To construct the grouped frequency distribution, frequency polygon, and ogive, we first need to organize the given data.

• Minimum value = 3
• Maximum value = 254
• Range = Maximum value - Minimum value = $254 - 3 = 251$

Using Sturges' formula for the number of classes ($k$):

$$k = 1 + 3.322 \log_{10} N$$ $$k = 1 + 3.322 \log_{10} 50$$ $$k = 1 + 3.322 \times 1.69897$$ $$k \approx 1 + 5.64 = 6.64$$

We choose $k=7$ classes.

Class width ($w$) = $\frac{\text{Range}}{k} = \frac{251}{7} \approx 35.85$. We round up to a convenient number, $w=40$.

We define the class intervals starting from a value slightly below the minimum (or the minimum itself) and use a class width of 40.

Class Interval	Class Boundaries	Midpoint ($x$)	Frequency ($f$)	Cumulative Frequency ($cf$)
0-39	-0.5 - 39.5	19.5	15	15
40-79	39.5 - 79.5	59.5	16	31
80-119	79.5 - 119.5	99.5	10	41
120-159	119.5 - 159.5	139.5	3	44
160-199	159.5 - 199.5	179.5	3	47
200-239	199.5 - 239.5	219.5	2	49
240-279	239.5 - 279.5	259.5	1	50
Total			50

A frequency polygon is constructed by plotting the midpoint of each class against its frequency and connecting the points with straight lines. To close the polygon, add a class with zero frequency at each end.

The points to plot are:

• (-20.5, 0) (midpoint of a hypothetical class before 0-39)
• (19.5, 15)
• (59.5, 16)
• (99.5, 10)
• (139.5, 3)
• (179.5, 3)
• (219.5, 2)
• (259.5, 1)
• (299.5, 0) (midpoint of a hypothetical class after 240-279)

An ogive is constructed by plotting the upper class boundary of each class against its cumulative frequency and connecting the points with straight lines.

The points to plot are:

• (-0.5, 0) (lower class boundary of the first class with cumulative frequency 0)
• (39.5, 15)
• (79.5, 31)
• (119.5, 41)
• (159.5, 44)
• (199.5, 47)
• (239.5, 49)
• (279.5, 50)