Chapter 9Chapter 9StatisticsStatistics

Section 9.1Section 9.1

Frequency Distributions;Frequency Distributions;

Measures of Central TendencyMeasures of Central Tendency

Random SamplesRandom Samples

When a characteristic of a population When a characteristic of a population needs to be studied, it is sometimes needs to be studied, it is sometimes not possible to examine all the not possible to examine all the elements in the population.elements in the population.

A limited sample is used when the A limited sample is used when the population is too large.population is too large.

In order for the inferences gained In order for the inferences gained from the study to be correct, the from the study to be correct, the sample chosen must be a sample chosen must be a random random samplesample..

Random SamplesRandom Samples

Random samples are representative Random samples are representative of the population because they are of the population because they are chosen so that every element of the chosen so that every element of the population is equally likely to be population is equally likely to be selected.selected.

Often difficult to obtain in real life.Often difficult to obtain in real life. Once a sample has been chosen and Once a sample has been chosen and

all data collected, the data must be all data collected, the data must be organized so that conclusions may be organized so that conclusions may be more easily drawn.more easily drawn.

Organizing DataOrganizing Data

One method of organizing data is to One method of organizing data is to group the data into intervals (usually group the data into intervals (usually equal intervals).equal intervals).

A A grouped frequency distributiongrouped frequency distribution is a is a table that displays each interval and table that displays each interval and the number of times data points the number of times data points occur in the intervals.occur in the intervals.

Recommendations for the Number Recommendations for the Number of College Units in Management a of College Units in Management a

Business Major Should HaveBusiness Major Should Have

Graphically Organizing Data Graphically Organizing Data

The information in a grouped The information in a grouped frequency distribution can be frequency distribution can be displayed in a histogram similar to the displayed in a histogram similar to the histograms for probability distributions. histograms for probability distributions.

The intervals determine the widths of The intervals determine the widths of the bars. (Equal intervals = equal bar the bars. (Equal intervals = equal bar widths)widths)

The heights of the bars are determined The heights of the bars are determined by the frequencies.by the frequencies.

Frequency PolygonsFrequency Polygons

A A frequency polygonfrequency polygon is another form is another form of graph that illustrates a grouped of graph that illustrates a grouped frequency distribution.frequency distribution.

The polygon is formed by joining The polygon is formed by joining consecutive midpoints of the tops of consecutive midpoints of the tops of the histogram bars with straight line the histogram bars with straight line segments. segments.

The midpoints of the first and last The midpoints of the first and last bars are joined to endpoints on the bars are joined to endpoints on the horizontal axis where the next horizontal axis where the next midpoint would appear.midpoint would appear.

Recommendations for the Number of Recommendations for the Number of College Units in Management a College Units in Management a

Business Major Should HaveBusiness Major Should Have

Example 1Example 1 Trooper Barney Fife recorded the following Trooper Barney Fife recorded the following

speeds along Mayberry Highway during a 1 speeds along Mayberry Highway during a 1 hour period. All speeds are mph.hour period. All speeds are mph.

4545 3939 5555 4242 5252

3939 4646 6161 5454 3333

5050 4747 2929 5757 60 60

Construct a frequency distribution, Construct a frequency distribution, histogram, and histogram, and

frequency polygon for this information.frequency polygon for this information.

Measures of Central Measures of Central TendencyTendency

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

MeanMean

MedianMedian

ModeMode

MeanMean Most important, and commonly used, measure Most important, and commonly used, measure

of central tendency.of central tendency. The The arithmetic meanarithmetic mean (the (the meanmean) of a set of ) of a set of

numbers is the sum of the numbers, divided by numbers is the sum of the numbers, divided by the total number of numbers.the total number of numbers.

Example 2Example 2

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

4545 3939 5555 4242 5252

3939 4646 6161 5454 3333

5050 4747 2929 5757 60 60

Mean for Frequency Mean for Frequency DistributionsDistributions

Example 3Example 3

Use the frequency distribution for Use the frequency distribution for Trooper Fife’s data to determine the Trooper Fife’s data to determine the mean.mean.

Compare this mean to the mean of Compare this mean to the mean of the ungrouped data.the ungrouped data.

Two Types of MeanTwo Types of Mean

Sample meanSample mean: mean of a random sample. : mean of a random sample. This mean is used most often when the This mean is used most often when the population is very large.population is very large.

Population meanPopulation mean: mean for the entire : mean for the entire population. The expected value of a population. The expected value of a random variable in a probability random variable in a probability distribution is sometimes called the distribution is sometimes called the population mean. Denoted by µ. (Greek population mean. Denoted by µ. (Greek letter “mu”)letter “mu”)

MedianMedian

The The medianmedian is the middle entry in a is the middle entry in a set of data arranged in either set of data arranged in either increasing or decreasing order.increasing or decreasing order.

If there is an even number of entries, If there is an even number of entries, the median is defined to be the mean the median is defined to be the mean of the two center entries.of the two center entries.

StatisticStatistic

Both the mean and median are Both the mean and median are examples of a examples of a statisticstatistic, which is , which is simply a number that gives simply a number that gives information about a sample.information about a sample.

Sometimes the median gives a truer Sometimes the median gives a truer representation or typical element of representation or typical element of the data than the mean.the data than the mean.

The mean is sometimes not the best The mean is sometimes not the best representation because it is easily representation because it is easily influenced by extreme outliers.influenced by extreme outliers.

ModeMode

The The modemode is the most frequently is the most frequently occurring entry, or entries, in a set of occurring entry, or entries, in a set of data.data.

If there is no one entry that occurs If there is no one entry that occurs more than the others we say there is more than the others we say there is no modeno mode..

Sometimes, the data set will have Sometimes, the data set will have more than one mode.more than one mode.

Example 4Example 4

Find the speed along Mayberry HighwayFind the speed along Mayberry Highway

that would represent the median speed.that would represent the median speed.

What speed(s) represent the mode, if What speed(s) represent the mode, if any?any?

4545 3939 5555 4242 5252

3939 4646 6161 5454 3333

5050 4747 2929 5757 60 60

Advantages and Disadvantages for Advantages and Disadvantages for Mean, Median, and ModeMean, Median, and Mode

Measure Measure of Central of Central TendencyTendency

AdvantageAdvantage DisadvantageDisadvantage

MeanMean

Easy to computeEasy to computeTakes all data values into Takes all data values into accountaccountReliableReliable

Influenced by Influenced by extreme valuesextreme values

MedianMedian

Easy to computeEasy to computeNot influenced by extreme Not influenced by extreme valuesvaluesCan be used with non-Can be used with non-numerical datanumerical data

Difficult to rank Difficult to rank large number of large number of data valuesdata values

ModeMode

Easily foundEasily foundNot influenced by extreme Not influenced by extreme valuesvaluesCan be used with non-Can be used with non-numerical datanumerical data

Can’t always locate Can’t always locate just one modejust one mode