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The mean, often referred to as the average, is a fundamental concept in descriptive statistics that reflects the central tendency of a dataset. To calculate the mean, you sum up all the numbers in your dataset and then divide by the total number of values. In the case of DVRs in households, this entails adding up all the DVR counts and dividing by 25 (the total number of households in the sample).
The formula for the mean is given by:

• Mean = \( \frac{\text{Sum of all values}}{\text{Number of values}} \)
• For our example, Mean = \( \frac{71}{25} = 2.84 \)

This mean value tells us that, on average, each household in the sample has about 2.84 DVRs. The mean is sensitive to outliers, meaning that unusually high or low values can skew the result.

The median represents the midpoint of a dataset, where half of the numbers are above this value, and half are below it. To find the median, first order the data from smallest to largest. In a dataset with an odd number of observations, as is the case with our 25 households, the median is simply the value in the middle position.
For the DVR dataset:

• Arrange data: (1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5)
• The 13th value = 2, which is the median.

The median is not affected by outliers or skewed data as much as the mean, which makes it a more reliable measure when a dataset includes extremely high or low values. It clearly shows that half of the households have more than two DVRs, and half have less or exactly two.

The mode is the value that appears most frequently in a dataset. It is particularly useful in understanding which value is most common in a set of observations. In the example of DVRs in households, the mode can be determined by counting the frequency of each value and selecting the one that occurs the most often.
From the given data:

• DVR count: (1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5)
• The value 2 occurs 9 times, making it the mode.

The mode helps in identifying the most typical DVR count among households, which in this case is 2. Unlike the mean and median, a dataset can have more than one mode, or no mode, depending on the distribution of values.

A relative frequency histogram is an easy-to-understand graphical representation showing how frequently different values occur in a dataset, scaled by the total number of observations. Each bar in the histogram represents a "relative frequency," which is the ratio of the frequency of a particular value to the total number of values. In this context, it helps us visualize how DVRs are distributed across households.
To construct the histogram:

• Calculate relative frequencies using the formula: \( \text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Number of Observations}} \)
• Example: For DVR count 1, Relative Frequency = \( \frac{4}{25} = 0.16 \)

Build the histogram with DVR numbers on the horizontal axis and their relative frequencies on the vertical axis.
This visualization will make it clear which counts are more common and how they compare with the mean, median, and mode—providing a comprehensive overview of the data's distribution.