Question

A survey asking for the number of times toast is burned during one week was distributed to eight randomly selected households. The survey yielded the following results $$ 2,3,0,3,4,1,3,0 $$ What is the range, variance and standard deviation for this data set?

Short answer

The range of the data set is 4, the variance is 2, and the standard deviation is approximately 1.41.

Step-by-step solution

Define and Sort the Dataset

The given dataset contains the number of times toast was burned in one week in eight randomly selected households. The data is as follows: \(2, 3, 0, 3, 4, 1, 3, 0\) Firstly, for convenience, list the numbers in the dataset in increasing order: \(0, 0, 1, 2, 3, 3, 3, 4\)

Calculate the Range

The range is the difference between the largest number and the smallest number in a dataset. In our case, the largest number (Maximum) is 4, and the smallest number (Minimum) is 0. So, the range would be: Range = Maximum - Minimum = 4 - 0 = 4

Calculate the Mean

The mean (or average) is calculated by adding up all the numbers in the dataset and dividing by the number of values in the dataset. Mean = (0 + 0 + 1 + 2 + 3 + 3 + 3 + 4) / 8 = 2

Calculate the Variance

Variance is the average of the squared differences from the mean. Here's how to do it step by step: 1. Subtract the mean from each number (data point) and square the result. 2. Then calculate the average of those squared differences. - \((0 - 2)^2 = 4\) - \((0 - 2)^2 = 4\) - \((1 - 2)^2 = 1\) - \((2 - 2)^2 = 0\) - \((3 - 2)^2 = 1\) - \((3 - 2)^2 = 1\) - \((3 - 2)^2 = 1\) - \((4 - 2)^2 = 4\) Mean of squared differences (Variance) = \((4 + 4 + 1 + 0 + 1 + 1 + 1 + 4) / 8 = \frac{16}{8}=2\)

Calculate the Standard Deviation

The standard deviation is the square root of the variance. Standard Deviation = \(\sqrt{Variance} = \sqrt{2} \approx 1.41\) So, the range, variance and standard deviation for this data set are 4, 2, and 1.41 respectively.

Key concepts

Range

The range of a data set is a measure of how spread out the values are. It is calculated simply by subtracting the smallest value from the largest value. In our example, the range is calculated by taking the highest number of burned toast instances (4) and subtracting the lowest number of instances (0), which gives us a range of 4. This tells us that the spread of burned toast occurrences over one week varies by up to 4 times between the households surveyed.

Understanding the range can give us a quick glimpse into the variability of a data set, but it doesn't tell us much about the distribution pattern of all values within the dataset.

Variance

Variance provides a more nuanced picture of the spread than the range. It tells us how much the data points, in a set, deviate from the mean (average) value. This calculation might seem complicated, but it's all about finding the mean of the squared differences from the mean of the dataset.

To calculate the variance, as seen in our example, we first find the difference between each data point and the mean, square that difference, and then work out the average of those squared differences. A higher variance indicates that data points fall further away from the mean, showing a wider spread of the data. In our toast burning scenario, the variance was 2, indicating that the frequency of burning toast varies somewhat from household to household.

Standard Deviation

The standard deviation is the square root of the variance and offers another perspective on the spread of a data set. Unlike variance, which is squared, the standard deviation is in the same units as the data, making it more interpretable. For example, with a standard deviation of approximately 1.41, we can say that the number of times toast is burned in a week typically varies by about 1.41 from the average number in our set of surveyed households.

Standard deviation is widely used because it is directly relatable to the data points. It tells us, on average, how far each data point is from the mean and thus is crucial for understanding the reliability of the mean as a measure of central tendency.

Mean

The mean, commonly known as the average, is a central measure of a data set, calculated by adding all the values together and then dividing by the number of values. In our exercise, after summing up the instances of burned toast (0, 0, 1, 2, 3, 3, 3, 4) and dividing by the number of households (8), we found the mean to be 2.

The mean gives us a central point to compare each value in the set against and is used to calculate other important statistics like the variance and standard deviation. However, it's important to note that the mean can sometimes be misleading if the data set has outliers, as it can be skewed by the unusually high or low values.

Data Set Analysis

Data set analysis involves using measures of central tendency and variability, like mean, range, variance, and standard deviation, to summarize and describe the features of a collection of data. By analyzing the survey results from the eight households regarding their burned toast incidents, we've been able to gain insights not just into the average number of burnings but also into the consistency and spread of the burnings across different households.

Understanding these measures allows us to interpret various aspects of data sets effectively. For instance, even if two sets of data have the same mean, their variance and standard deviation can tell a different story about their spread and, consequently, about their reliability or predictability. Such analysis is essential in fields like quality control, economics, and social sciences, where making sense of data is fundamental to drawing meaningful conclusions.