Chapter Three Numerically Summarizing Data 3.1 Measures of Central Tendency

A parameter is a descriptive measure of a population. A statistic is a descriptive measure of a sample. A statistic is an unbiased estimator of a parameter if it does not consistently over- or underestimate the parameter.

The arithmetic mean of a variable is computed by determining the sum of all the values of the variable in the data set divided by the number of observations.

The population arithmetic mean, is computed using all the individuals in a population. The population mean is a parameter. The population arithmetic is denoted by

The sample arithmetic mean, is computed using sample data. The sample mean is a statistic that is an unbiased estimator of the population mean. The sample arithmetic is denoted by

Compute the population mean of this data. EXAMPLE Computing a Population Mean and a Sample Mean Treat the students in class as a population. Have the students provide some data for some quantitative variable such as pulse or number of siblings. Compute the population mean of this data. Then take a simple random sample of n = 5 students. Compute the sample mean. Obtain a second simple random sample of n = 5 students. Again compute the sample mean.

The median of a variable is the value that lies in the middle of the data when arranged in ascending order. That is, half the data is below the median and half the data is above the median. We use M to represent the median.

EXAMPLE Computing the Median of Data Find the median of the population data from the earlier example.

The mode of a variable is the most frequent observation of the variable that occurs in the data set. If there is no observation that occurs with the most frequency, we say the data has no mode.

EXAMPLE Finding the Mode of a Data Set The data on the next slide represent the Vice Presidents of the United States and their state of birth. Find the mode.

The mode is New York.

The arithmetic mean is sensitive to extreme (very large or small) values in the data set, while the median is not. We say the median is resistant to extreme values, but the arithmetic mean is not.

When data sets have unusually large or small values relative to the entire set of data or when the distribution of the data is skewed, the median is the preferred measure of central tendency over the arithmetic mean because it is more representative of the typical observation.

EXAMPLE. Identifying the Shape of the Distribution EXAMPLE Identifying the Shape of the Distribution Based on the Mean and Median The following data represent the asking price of homes for sale in Lincoln, NE. Source: http://www.homeseekers.com

Find the mean and median Find the mean and median. Use the mean and median to identify the shape of the distribution. Verify your result by drawing a histogram of the data.

Find the mean and median Find the mean and median. Use the mean and median to identify the shape of the distribution. Verify your result by drawing a histogram of the data. Using MINITAB, we find that the mean asking price is $143,509 and the median asking price is $131,825. Therefore, we would conjecture that the distribution is skewed right.