Statistics question and answer

All QuestionsCategory: Secondary SchoolStatistics question and answer
Bos asked 1 year ago

Below are the marks scored by 60 students in mathematics:72,72,57,65,68,70,79,53,68,70,70,62,64,82,74,75,74,62,70,91,74,80,72,76,79,65,67,84,91,61,73,70,58,70,56,70,56,71,80,79,61,80,76,65,78,92,66,86,84,77,80,69,72,62,76,67,74,63,86,73,69Use the above data with the given interval 50-54,55-59,60-64 and construct a cumulative frequency distibution table,draw a cumulative frequency curve,from your curve,determine the following: Median Inter quartile range 30th percentile.

2 Answers
User AvatarStopLearn Team Staff answered 1 year ago

To construct a cumulative frequency distribution table, we first need to determine the class intervals and calculate the cumulative frequency. The given intervals are:
50-54, 55-59, 60-64, 65-69, 70-74, 75-79, 80-84, 85-89, 90-94
The frequency count for each interval is:
50-54: 0 55-59: 2 60-64: 7 65-69: 9 70-74: 16 75-79: 9 80-84: 8 85-89: 2 90-94: 1
To calculate the cumulative frequency, we add up the frequency of each interval and the frequencies of all the previous intervals:
50-54: 0 55-59: 2 60-64: 9 (7+2) 65-69: 18 (9+9) 70-74: 34 (16+18) 75-79: 43 (9+34) 80-84: 51 (8+43) 85-89: 53 (2+51) 90-94: 54 (1+53)
The cumulative frequency distribution table looks like this:

Interval
Frequency
Cumulative Frequency

50-54
0
0

55-59
2
2

60-64
7
9

65-69
9
18

70-74
16
34

75-79
9
43

80-84
8
51

85-89
2
53

90-94
1
54

To draw the cumulative frequency curve, we plot the cumulative frequency against the upper limit of each interval.
To find the median, we look for the value that splits the distribution into two equal halves. Since there are 60 observations, the median is the average of the 30th and 31st observations when the data is arranged in ascending order. We can see from the cumulative frequency curve that the 30th and 31st observations fall in the interval 70-74. The lower limit of this interval is 70, and the width of the interval is 5. The cumulative frequency up to this interval is 34, and the frequency of this interval is 16. So, the median can be calculated as:
Median = lower limit of the median interval + [(n/2 – cumulative frequency of the previous interval) / frequency of the median interval] x width of the interval = 70 + [(30 – 34) / 16] x 5 = 70 + (-0.25) x 5 = 68.75
Therefore, the median is 68.75.
To find the interquartile range (IQR), we need to first find the quartiles Q1 and Q3. The 25th percentile (Q1) is the value that separates the bottom 25% of the data from the top 75%, and the 75th percentile (Q3) is the value that separates the bottom 75% of the data from the top 25%. We can find these values using the cumulative frequency curve.

User AvatarStopLearn Team Staff answered 1 year ago

To construct a cumulative frequency distribution table, we first need to determine the class intervals and calculate the cumulative frequency. The given intervals are:
50-54, 55-59, 60-64, 65-69, 70-74, 75-79, 80-84, 85-89, 90-94
The frequency count for each interval is:
50-54: 0 55-59: 2 60-64: 7 65-69: 9 70-74: 16 75-79: 9 80-84: 8 85-89: 2 90-94: 1
To calculate the cumulative frequency, we add up the frequency of each interval and the frequencies of all the previous intervals:
50-54: 0 55-59: 2 60-64: 9 (7+2) 65-69: 18 (9+9) 70-74: 34 (16+18) 75-79: 43 (9+34) 80-84: 51 (8+43) 85-89: 53 (2+51) 90-94: 54 (1+53)
The cumulative frequency distribution table looks like this:

Interval
Frequency
Cumulative Frequency

50-54
0
0

55-59
2
2

60-64
7
9

65-69
9
18

70-74
16
34

75-79
9
43

80-84
8
51

85-89
2
53

90-94
1
54

To draw the cumulative frequency curve, we plot the cumulative frequency against the upper limit of each interval.
 
To find the median, we look for the value that splits the distribution into two equal halves. Since there are 60 observations, the median is the average of the 30th and 31st observations when the data is arranged in ascending order. We can see from the cumulative frequency curve that the 30th and 31st observations fall in the interval 70-74. The lower limit of this interval is 70, and the width of the interval is 5. The cumulative frequency up to this interval is 34, and the frequency of this interval is 16. So, the median can be calculated as:
Median = lower limit of the median interval + [(n/2 – cumulative frequency of the previous interval) / frequency of the median interval] x width of the interval
= 70 + [(30 – 34) / 16] x 5
= 70 + (-0.25) x 5
= 68.75
Therefore, the median is 68.75.
To find the interquartile range (IQR), we need to first find the quartiles Q1 and Q3. The 25th percentile (Q1) is the value that separates the bottom 25% of the data from the top 75%, and the 75th percentile (Q3) is the value that separates the bottom 75% of the data from the top 25%. We can find these values using the cumulative frequency curve.

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