The problem provides a dataset of 40 systolic blood pressure measurements. The task is to calculate the following statistical measures: mean, median, mode, standard deviation, variance, coefficient of variation, and Pearson's measure of skewness.

1. Problem Description

The problem provides a dataset of 40 systolic blood pressure measurements. The task is 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, input the data into a list.

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

I. Mean: The mean is the average of the data.

Formula:

$Mean = \frac{\sum_{i=1}^{n} x_i}{n}$

where $x_i$ are the data points and $n$ is the number of data points.

```python

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

n = len(data)

mean = sum(data) / n

print(mean)

```

Mean = 118.6

II. Median: The median is the middle value when the data is sorted.

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

```python

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

data.sort()

median = (data[19] + data[20]) / 2

print(median)

```

Median = (118 + 118) / 2 = 118

III. Mode: The mode is the value that appears most frequently in the data.

```python

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

counts = {}

for x in data:

if x in counts:

counts[x] += 1

else:

counts[x] = 1

mode = max(counts, key=counts.get)

print(mode)

```

The number 117 appears four times, more than any other number.

Mode = 117

IV. Standard Deviation:

Formula:

$s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}$

where $x_i$ are the data points, $\bar{x}$ is the mean, and $n$ is the number of data points.

```python

import math

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

n = len(data)

mean = sum(data) / n

variance = sum([(x - mean) ** 2 for x in data]) / (n - 1)

std_dev = math.sqrt(variance)

print(std_dev)

```

Standard Deviation = 7.76

V. Variance: The variance is the square of the standard deviation.

Formula:

$s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$

where $x_i$ are the data points, $\bar{x}$ is the mean, and $n$ is the number of data points.

```python

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

n = len(data)

mean = sum(data) / n

variance = sum([(x - mean) ** 2 for x in data]) / (n - 1)

print(variance)

```

Variance = 60.205128205128205

VI. Coefficient of Variation:

Formula:

$CV = \frac{s}{\bar{x}}$

where $s$ is the standard deviation and $\bar{x}$ is the mean.

```python

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

import math

n = len(data)

mean = sum(data) / n

variance = sum([(x - mean) ** 2 for x in data]) / (n - 1)

std_dev = math.sqrt(variance)

cv = std_dev / mean

print(cv)

```

Coefficient of Variation = 0.065434

VII. Pearson's Measure of Skewness:

Formula:

$Skewness = \frac{3(\bar{x} - Median)}{s}$

where $\bar{x}$ is the mean, $Median$ is the median, and $s$ is the standard deviation.

```python

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

import math

n = len(data)

mean = sum(data) / n

data.sort()

median = (data[19] + data[20]) / 2

variance = sum([(x - mean) ** 2 for x in data]) / (n - 1)

std_dev = math.sqrt(variance)

skewness = 3 * (mean - median) / std_dev

print(skewness)

```

Pearson's Measure of Skewness = 0.0

3. Final Answer

I. Mean = 118.6

II. Median = 118

III. Mode = 117

IV. Standard Deviation = 7.76

V. Variance = 60.21

VI. Coefficient of Variation = 0.065

VII. Pearson's Measure of Skewness = 0.0