Question

You are given \(n=8\) measurements: \(4,1,3,1,3,\) 1,2,2 a. Find the range. b. Calculate \(\bar{x}\). c. Calculate \(s^{2}\) and \(s\) using the computing formula. d. Use the data entry method in your calculator to find \(\bar{x}, s,\) and \(s^{2} .\) Verify that your answers are the same as those in parts \(\mathrm{b}\) and \(\mathrm{c}\).

Short answer

Question: Calculate the range, mean, sample variance, and standard deviation for the following data set: 1, 1, 1, 2, 2, 3, 3, 4. Also, verify the obtained results using a calculator.

Step-by-step solution

Rearrange the measurements in ascending order

It's helpful to rewrite the data in ascending order so we can easily identify the largest and smallest values: 1,1,1,2,2,3,3,4

Find the range

The range is the difference between the highest and lowest values. In this case, the highest value is 4, and the lowest value is 1. So, the range is \(4-1 = 3\).

Calculate the mean \(\bar{x}\)

To find the mean \(\bar{x}\), add up all the values and divide the sum by the number of values (n). In this case, \(\bar{x} = \frac{1+1+1+2+2+3+3+4}{8} = \frac{17}{8} = 2.125\).

Calculate the sample variance \(s^2\) using the computing formula

The computing formula for the sample variance is \(s^2 = \frac{\sum{x_i^2} - \frac{(\sum{x_i})^2}{n}}{n-1}\). First, calculate the sum of the squared values: \(1^2+1^2+1^2+2^2+2^2+3^2+3^2+4^2 = 1+1+1+4+4+9+9+16 = 45\) Now, calculate \(\frac{(\sum{x_i})^2}{n}\): \(\frac{(1+1+1+2+2+3+3+4)^2}{8} = \frac{17^2}{8} = \frac{289}{8} = 36.125\) Next, apply the computing formula for the sample variance: \(s^2 = \frac{45 - 36.125}{8-1} = \frac{8.875}{7} = 1.26786\)

Calculate the sample standard deviation \(s\)

The sample standard deviation is the square root of the sample variance: \(s = \sqrt{s^2} = \sqrt{1.26786} = 1.126\)

Verify results using a calculator

Input the data set into the calculator using its data entry method to calculate the mean \(\bar{x}\), standard deviation \(s\), and variance \(s^2\). Compare these values to the ones calculated in the previous steps. They should be the same, confirming our calculations.

Key concepts

Sample Variance

In statistics, sample variance is a measure of the spread, or dispersion, of a set of data points. To put it simply, it tells us how much the data points in a sample deviate from the sample mean. Sample variance is an important concept because it provides us with an idea of how varied or consistent the data is.

When we calculate sample variance, it is vital to understand that we are working with a subset of the entire population for which we may want to make inferences. That's why we divide by the number of observations minus one (-1) when calculating it. This is to account for what we call Bessel's correction, which provides an unbiased estimate of the population variance.

To compute the sample variance (), first, we calculate the mean (also known as the average) of the data points. Then, we find the difference of each data point from the mean and square these differences. After summing up all the squared differences, we divide by the total number of data points minus one. This can be expressed by the formula: . In the given exercise, following this method led to the calculation of a sample variance of 1.26786, which means the data points in the set have a moderate amount of variability around the mean.

Range Calculation

The range of a data set gives us a quick sense of how spread out the measurements are. It is the simplest measure of dispersion. To find the range, you simply subtract the smallest value from the largest value in your dataset.

This calculation provides us with the span of our data set but doesn't tell us anything about how the data is distributed within that range. Also, the range can be heavily influenced by extreme values or outliers in your data.

Despite this limitation, the range is often used because it is easy to calculate and understand. In the given problem, the range is quickly found by subtracting the smallest observation, 1, from the largest observation, 4, which results in a range of 3.

Mean Computation

The mean, often referred to as the average, is a fundamental statistical measurement. It is calculated by summing up all the numerical values in a dataset and then dividing that total by the count of the values.

The mean is a form of a measure of central tendency, which gives us a sense of the 'center' of the data. However, it is important to note that the mean can be affected by outliers and may not always represent the 'typical' value, especially in skewed distributions.

In our example, the mean computation involves adding together the eight numbers and dividing by eight, resulting in a mean of 2.125. This represents the central value around which the other data points in our sample are distributed.

Standard Deviation

The standard deviation is a widely used measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean, while a high standard deviation means that the values are spread out over a wider range.

The standard deviation is the square root of the variance. It is used instead of variance to match the units of the mean and the data themselves. It's particularly useful because it is expressed in the same units as the data, which makes it easier to understand in the context of the data that we are analyzing.

Using the data from the problem, the standard deviation was calculated to be 1.126. This provides us with a quantitative measure of how each measurement varies from the average of the measurements.