Question

The following data represent the Intelligence Quotient Scores of a random sample of thirty adults.  Construct a Stem-and-Leaf Plot AND a Five Number Summary for the measurements.  Compute the locator of the median, the median, AND the mean of the thirty measurements. Sample of Size n = 30 108 100 103 125 108 90 122 89 96 99 100 84 121 129 91 103 90 106 75 98 102 89 99 82 90 100 120 114 93 104

Answer 1

A stem and leaf plot shows numerical data by splitting each data point into a "leaf" (generally the unit digit) and a "stem" (the leading digit or digits).

Here, n=30

Here is the dataset of the Intelligence Quotient scores of a random sample of 30 adults and the stem and leaf plot:

Dataset :108 100 103 125 108 90 122 89 96 99 100 84 121 129 91 103 90 106 75 98 102 89 99 82 90 100 120 114 93 104.

Here, the scores of intelligence Quotient are ranging between 75-129.

So the smallest stem is 7 and largest stem is 12 for this data set.

Stem and leaf plot :

7 | 5

8 | 2, 4, 9, 9

9 | 0, 0, 0, 1, 3, 6, 8, 9, 9

10 | 0, 0, 0, 2, 3, 3, 4, 6, 8, 8

11 | 4

12 | 0, 1, 2, 5, 9

As we know five number summary contains values of minimum, lower quartile(L.Q), median, upper quartile(U.Q), maximum of given data.

By observing given dataset we get minimum of dataset is 75 and maximum is 129.

Median is the middle value of the ascending order arranged dataset.

Lower quartile of a stem and leaf data is the value of the median of the lower half scores of stem and leaf chart.

Upper quartile of a stem and leaf data is the value of the median of the upper half scores of the stem and leaf chart.

By applying above information we calculate the values of five number summary.

Five number summary:

Minimum - 75

Lower quartile - 90

Median - 100

Upper quartile - 108

Maximum - 129.

We know formula for median when sample size is even is

Median = [X(n/2) + X(n/2)+1]/ 2

Therefore median locator formula is (X15  +X16)/2.

Median : It is the value of middle most observation in the ascending or descending order arranged data of their values.

Median = 100

Mean = sum of all given observations/ number of observations .

Mean= ( 108 +100 +103 +125 +108 + 90 + 122 + 89 + 96 + 99 +100 + 84 + 121 + 129 + 91 + 103 + 90 + 106 + 75 + 98 + 102 + 89 + 99 + 82 + 90 + 100 + 120 + 114 + 93 + 104 )/30

Mean = 3030/30

Mean =101.