Using the following array of data, find the mean,mode and median for group data
Class interval : 20-25, 26-30, 31-35, 36-40, 41-45, 46-50, 51-55, 56-60, 61-65, 66-70.
Frequency : 22, 26, 42, 32, 38, 66, 36, 22, 18, 20.
To find the mean, mode, and median for grouped data, we need to use the midpoint of each class interval to represent the data.
Midpoints:
22.5, 28, 33, 38, 43, 48, 53, 58, 63, 68
To find the mean, we multiply each midpoint by its frequency, sum these products, and then divide by the total frequency.
Mean = (22.5*22 + 28*26 + 33*42 + 38*32 + 43*38 + 48*66 + 53*36 + 58*22 + 63*18 + 68*20) / 304
Mean = 3525 / 304
Mean = 11.60
The mean for the grouped data is 41.60.
To find the mode, we look for the class interval with the highest frequency. In this case, the interval 46-50 has the highest frequency of 66.
The mode for the grouped data is 46-50.
To find the median, we first calculate the position of the median within the data set. The median position is equal to (n+1) / 2, where n is the total frequency.
Median position = (304 + 1) / 2
Median position = 152.5
The median falls within the 51-55 class interval. To find the exact median, we need to use the cumulative frequency.
Cumulative frequency:
22, 48, 90, 122, 160, 226, 262, 284, 302, 322
The median falls within the 51-55 class interval, so we can use the formula:
Median = L + (n/2 - F) * w / f
where:
L = lower boundary of the median class interval (51)
n = total frequency (304)
F = cumulative frequency of the interval preceding the median class interval (160)
w = width of the class interval (5)
f = frequency of the median class interval (36)
Median = 51 + (152.5 - 160) * 5 / 36
Median = 51 + (-7.5) * 5 / 36
Median = 51 - 37.5 / 36
Median = 51 - 1.04
The median for the grouped data is 49.96.