Chapter 9 Statistics Section 9.1 Frequency Distributions; Measures of Central Tendency

Random Samples When a characteristic of a population needs to be studied, it is sometimes not possible to examine all the elements in the population. When a characteristic of a population needs to be studied, it is sometimes not possible to examine all the elements in the population. A limited sample is used when the population is too large. A limited sample is used when the population is too large. In order for the inferences gained from the study to be correct, the sample chosen must be a random sample. In order for the inferences gained from the study to be correct, the sample chosen must be a random sample.

Random Samples Random samples are representative of the population because they are chosen so that every element of the population is equally likely to be selected. Random samples are representative of the population because they are chosen so that every element of the population is equally likely to be selected. Often difficult to obtain in real life. Often difficult to obtain in real life. Once a sample has been chosen and all data collected, the data must be organized so that conclusions may be more easily drawn. Once a sample has been chosen and all data collected, the data must be organized so that conclusions may be more easily drawn.

Organizing Data One method of organizing data is to group the data into intervals (usually equal intervals). One method of organizing data is to group the data into intervals (usually equal intervals). A grouped frequency distribution is a table that displays each interval and the number of times data points occur in the intervals. A grouped frequency distribution is a table that displays each interval and the number of times data points occur in the intervals.

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Graphically Organizing Data The information in a grouped frequency distribution can be displayed in a histogram similar to the histograms for probability distributions. The information in a grouped frequency distribution can be displayed in a histogram similar to the histograms for probability distributions. The intervals determine the widths of the bars. (Equal intervals = equal bar widths) The intervals determine the widths of the bars. (Equal intervals = equal bar widths) The heights of the bars are determined by the frequencies. The heights of the bars are determined by the frequencies.

Frequency Polygons A frequency polygon is another form of graph that illustrates a grouped frequency distribution. A frequency polygon is another form of graph that illustrates a grouped frequency distribution. The polygon is formed by joining consecutive midpoints of the tops of the histogram bars with straight line segments. The polygon is formed by joining consecutive midpoints of the tops of the histogram bars with straight line segments. The midpoints of the first and last bars are joined to endpoints on the horizontal axis where the next midpoint would appear. The midpoints of the first and last bars are joined to endpoints on the horizontal axis where the next midpoint would appear.

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Example 1 Trooper Barney Fife recorded the following speeds along Mayberry Highway during a 1 hour period. All speeds are mph. Trooper Barney Fife recorded the following speeds along Mayberry Highway during a 1 hour period. All speeds are mph Construct a frequency distribution, histogram, and frequency polygon for this information.

Measures of Central Tendency Three measures of central tendency, or “averages” are used with frequency distributions to describe the data. Three measures of central tendency, or “averages” are used with frequency distributions to describe the data.MeanMedianMode

Mean Most important, and commonly used, measure of central tendency. Most important, and commonly used, measure of central tendency. The arithmetic mean (the mean) of a set of numbers is the sum of the numbers, divided by the total number of numbers. The arithmetic mean (the mean) of a set of numbers is the sum of the numbers, divided by the total number of numbers.

Example 2 Help Trooper Fife find the average speed along Mayberry Highway during that particular hour

Mean for Frequency Distributions

Example 3 Use the frequency distribution for Trooper Fife’s data to determine the mean. Use the frequency distribution for Trooper Fife’s data to determine the mean. Compare this mean to the mean of the ungrouped data. Compare this mean to the mean of the ungrouped data.

Two Types of Mean Sample mean: mean of a random sample. This mean is used most often when the population is very large. Sample mean: mean of a random sample. This mean is used most often when the population is very large. Population mean: mean for the entire population. The expected value of a random variable in a probability distribution is sometimes called the population mean. Denoted by µ. (Greek letter “mu”) Population mean: mean for the entire population. The expected value of a random variable in a probability distribution is sometimes called the population mean. Denoted by µ. (Greek letter “mu”)

Median The median is the middle entry in a set of data arranged in either increasing or decreasing order. The median is the middle entry in a set of data arranged in either increasing or decreasing order. If there is an even number of entries, the median is defined to be the mean of the two center entries. If there is an even number of entries, the median is defined to be the mean of the two center entries.

Statistic Both the mean and median are examples of a statistic, which is simply a number that gives information about a sample. Both the mean and median are examples of a statistic, which is simply a number that gives information about a sample. Sometimes the median gives a truer representation or typical element of the data than the mean. Sometimes the median gives a truer representation or typical element of the data than the mean. The mean is sometimes not the best representation because it is easily influenced by extreme outliers. The mean is sometimes not the best representation because it is easily influenced by extreme outliers.

Mode The mode is the most frequently occurring entry, or entries, in a set of data. The mode is the most frequently occurring entry, or entries, in a set of data. If there is no one entry that occurs more than the others we say there is no mode. If there is no one entry that occurs more than the others we say there is no mode. Sometimes, the data set will have more than one mode. Sometimes, the data set will have more than one mode.

Example 4 Find the speed along Mayberry Highway that would represent the median speed. What speed(s) represent the mode, if any?

Advantages and Disadvantages for Mean, Median, and Mode Measure of Central Tendency Measure of Central TendencyAdvantageDisadvantage Mean Easy to compute Easy to compute Takes all data values into account Takes all data values into account Reliable Reliable Influenced by extreme values Influenced by extreme values Median Easy to compute Easy to compute Not influenced by extreme values Not influenced by extreme values Can be used with non-numerical data Can be used with non-numerical data Difficult to rank large number of data values Difficult to rank large number of data values Mode Easily found Easily found Not influenced by extreme values Not influenced by extreme values Can be used with non-numerical data Can be used with non-numerical data Can’t always locate just one mode Can’t always locate just one mode