Question

Shown is an ogive depicting the cumulative frequency of the average mathematics SAT scores by state. Use it to construct a histogram and a frequency polygon.

Answer 1

Compute the class boundaries, midpoints (Xm) and frequency by using the given Ogive.

 

The general formula for the class boundaries and midpoint are given below

Lowe Boundary = Lower Limit – 0.5

Upper Boundary = Upper Limit + 0.5

Xm = (Lower limit + Upper limit)/2

 

From the given Ogive curve, the class boundaries, midpoints (Xm) and frequencies are tabulated below:

Boundaries	Midpoint (Xm)	Cumulative Frequency	Frequency
468.5-495.5	482	6	6
495.5-522.5	509	21	21 – 6 = 15
522.5-549.5	536	31	31 – 21 = 10
549.5-576.5	563	38	38 – 31 = 7
576.5-603.5	590	44	44 = 38 = 6
603.5-630.5	617	50	50 – 44 = 6
Total		50

 

Use the steps and draw the histogram of Math SAT scores.

Step 1: Draw the horizontal line (x axis) and label it as Average Mathematics Score.

Step 2: Represent the class boundaries on the x axis.

Step 3: Draw the vertical line (y axis) and label it as Frequency.

Step 4: Using the frequencies as the heights, draw the vertical bars for each class.

 

The histogram of Math SAT score is shown below:

 

Observation:

From the above histogram, it is clear that the right side of the histogram is extended larger than the left side. Hence, the distribution seems to be positively skewed.

 

Use the steps and draw the frequency polygon of Math SAT scores.

Step 1: Draw the horizontal line (x axis) and label it as Average Mathematics Score.

Step 2: Represent the midpoint of each class on the x axis.

Step 3: Draw the vertical line (y axis) and label it as Frequency.

Step 4: Plot the points using midpoints and frequencies.

Step 5: Join the adjacent points with line segments.

Step 6: Extend the line at the beginning and end of the graph where the earlier and next midpoints would be located on the x axis.

 

The frequency polygon of Math SAT score is shown below: