Most American households have one digital video recorder...

Most American households have one digital video recorder (DVR), and many have more than one. A sample of 25 households produced the following measurements on $x$, the number of DVRs in the household: $\begin{array}{lllll}1 & 0 & 2 & 1 & 1 \\ 1 & 0 & 2 & 1 & 0 \\ 0 & 1 & 2 & 3 & 2 \\ 1 & 1 & 1 & 0 & 1 \\ 3 & 1 & 0 & 1 & 1\end{array}$ a. Is the distribution of $x$, the number of DVRs in a household, symmetric or skewed? Explain. b. Guess the value of the mode, the value of $x$ that occurs most frequently. c. Calculate the mean, the median, and the mode for these measurements. d. Draw a relative frequency histogram for the data. Locate the mean, the median, and the mode along the horizontal axis. Are your answers to parts a and b correct?

Most American households have one digital video recorder (DVR), and many have more than one. A sample of 25 households produced the following measurements on $x$, the number of DVRs in the household:
$\begin{array}{lllll}1 & 0 & 2 & 1 & 1 \\ 1 & 0 & 2 & 1 & 0 \\ 0 & 1 & 2 & 3 & 2 \\ 1 & 1 & 1 & 0 & 1 \\ 3 & 1 & 0 & 1 & 1\end{array}$
a. Is the distribution of $x$, the number of DVRs in a household, symmetric or skewed? Explain.
b. Guess the value of the mode, the value of $x$ that occurs most frequently.
c. Calculate the mean, the median, and the mode for these measurements.
d. Draw a relative frequency histogram for the data. Locate the mean, the median, and the mode along the horizontal axis. Are your answers to parts a and b correct?

Key Concepts

Skewness

Skewness refers to the asymmetry in the distribution of a data set. A symmetric distribution has equal tails on both sides, while a skewed distribution exerts a longer tail on one side, which can affect the relationship among the mean, median, and mode.

Mode

The mode is the value that appears most frequently in a data set. It is particularly useful for identifying the most common or typical value, especially in cases where the data are categorical or have a discrete distribution.

Relative Frequency Histogram

A relative frequency histogram is a graphical representation that displays the distribution of a data set by showing the proportion of observations falling within different intervals or bins. It is useful for visualizing the shape, central tendency, and spread of the data, as well as for comparing theoretical models to the observed data distribution.

Mean

The mean is the arithmetic average of a set of values, calculated by summing all the values and then dividing by the number of observations. It provides a measure of the center of the data but can be influenced by extreme values in skewed distributions.

Descriptive Statistics

Descriptive statistics is the branch of statistics that involves summarizing and describing the essential features of a data set. This includes central tendency measures, variability, and the overall shape of the distribution, which helps in understanding and conveying the structure of the data.

Median

The median is the middle value of an ordered data set. When the data are arranged in increasing order, the median divides the data into two equal halves, offering a robust indicator of central tendency that is less affected by outliers and skewed distributions.

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