We are given a dataset of 40 systolic blood pressure measurements in mmHg. We need to calculate the following statistical measures: Mean, Median, Mode, Standard Deviation, Variance, Coefficient of Variation, and Pearson's Measure of Skewness.

1. Problem Description

We are given a dataset of 40 systolic blood pressure measurements in mmHg. We need to calculate the following statistical measures: Mean, Median, Mode, Standard Deviation, Variance, Coefficient of Variation, and Pearson's Measure of Skewness.

2. Solution Steps

First, we need to organize the data in ascending order:

101, 103, 105, 109, 109, 110, 112, 113, 114, 114, 114, 115, 115, 116, 117, 117, 117, 117, 118, 118, 118, 119, 119, 120, 121, 121, 122, 123, 123, 124, 124, 125, 125, 126, 127, 127, 128, 129, 130, 131

I. Mean:

The mean is the sum of all values divided by the number of values.

$Mean = \frac{\sum x_i}{n}$

$Mean = \frac{101 + 103 + 105 + 109 + 109 + 110 + 112 + 113 + 114 + 114 + 114 + 115 + 115 + 116 + 117 + 117 + 117 + 117 + 118 + 118 + 118 + 119 + 119 + 120 + 121 + 121 + 122 + 123 + 123 + 124 + 124 + 125 + 125 + 126 + 127 + 127 + 128 + 129 + 130 + 131}{40}$

$Mean = \frac{4776}{40} = 119.4$

II. Median:

Since there are 40 data points (an even number), the median is the average of the 20th and 21st values in the ordered list.

The 20th value is

1

1

8. The 21st value is

1

1

8. $Median = \frac{118 + 118}{2} = 118$

III. Mode:

The mode is the value that appears most frequently in the dataset.

117 appears 4 times, which is the highest frequency.

$Mode = 117$

IV. Standard Deviation:

$Standard Deviation = \sqrt{\frac{\sum (x_i - Mean)^2}{n-1}}$

First, calculate the sum of squared differences:

$\sum (x_i - Mean)^2 = (101-119.4)^2 + (103-119.4)^2 + ... + (131-119.4)^2 = 18.4^2 + 16.4^2 + 14.4^2 + ... + 11.6^2 = 1575.4$

$Standard Deviation = \sqrt{\frac{1575.4}{40-1}} = \sqrt{\frac{1575.4}{39}} = \sqrt{40.39487} \approx 6.35569$

$Standard Deviation \approx 6.36$

V. Variance:

Variance is the square of the standard deviation.

$Variance = (Standard Deviation)^2 = (6.35569)^2 = 40.39487 \approx 40.39$

VI. Coefficient of Variation:

The coefficient of variation is the ratio of the standard deviation to the mean, expressed as a percentage.

$CV = \frac{Standard Deviation}{Mean} \times 100\%$

$CV = \frac{6.35569}{119.4} \times 100\% = 0.05323 \times 100\% \approx 5.32\%$

VII. Pearson's Measure of Skewness:

$Skewness = \frac{3(Mean - Median)}{Standard Deviation}$

$Skewness = \frac{3(119.4 - 118)}{6.35569} = \frac{3(1.4)}{6.35569} = \frac{4.2}{6.35569} \approx 0.6608$

$Skewness \approx 0.66$

3. Final Answer

I. Mean: 119.4

II. Median: 118

III. Mode: 117

IV. Standard Deviation: 6.36

V. Variance: 40.39

VI. Coefficient of Variation: 5.32%

VII. Pearson's Measure of Skewness: 0.66