The Frequency Distribution Table
Class Interval Frequency
10-19 1
20-28 4
29-37 6
38-46 9
47-55 11
56-64 19
The grouped mode is 56-64,
The grouped median is 71.75
The grouped mean is 47.75.
Determining the class intervals. The class size is given as 9, so we can start with the minimum score and increment by 9 to determine the upper limits of each interval.
Minimum score: 10
Upper limits of intervals: 19, 28, 37, 46, 55, 64
Counting the number of scores falling within each interval:
10-19: 1
20-28: 4
29-37: 6
38-46: 9
47-55: 11
56-64: 19
Using this information, we can create the frequency distribution table:
Class Interval Frequency
10-19 1
20-28 4
29-37 6
38-46 9
47-55 11
56-64 19
Grouped Mode:
The class interval with the highest frequency is 56-64, with a frequency of 19.
Grouped Median:
The median is the middle value when the scores are arranged in ascending order. Since we have 50 scores, the median will be the average of the 25th and 26th scores.
The 25th and 26th scores fall within the class interval 47-55. To find the grouped median, we need to interpolate within this interval.
Lower limit of the interval: 47
Upper limit of the interval: 55
Number of scores in the interval: 11
Position of the median within the interval: (25 + 26) / 2 = 25.5
Using the formula:
Grouped Median = Lower limit + [(Position of median - Cumulative frequency of previous interval) / Frequency of the interval] × Class size
Grouped Median = 47 + [(25.5 - 9) / 11] × 9
Grouped Median = 47 + (16.5 / 11) × 9
Grouped Median = 47 + 24.75
Grouped Median = 71.75
Grouped Mean:
Midpoint of 10-19: 14.5
Midpoint of 20-28: 24
Midpoint of 29-37: 33
Midpoint of 38-46: 42
Midpoint of 47-55: 51
Midpoint of 56-64: 60
Calculating the grouped mean:
Grouped Mean = (14.5×1 + 24×4 + 33×6 + 42×9 + 51×11 + 60×19) / 50
Grouped Mean = (14.5 + 96 + 198 + 378 + 561 + 1140) / 50
Grouped Mean = 2387.5 / 50
Grouped Mean = 47.75
Therefore, the grouped mode is 56-64, the grouped median is 71.75, and the grouped mean is 47.75.