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Chapter 9 Statistics

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.

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Chapter 9 Statistics

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

  2. 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. • 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.

  3. 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. • 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.

  4. Organizing Data • 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.

  5. Recommendations for the Number of College Units in Management a Business Major Should Have

  6. Graphically Organizing Data • 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 heights of the bars are determined by the frequencies.

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

  8. Recommendations for the Number of College Units in Management a Business Major Should Have

  9. Example 1 • Trooper Barney Fife recorded the following speeds along Mayberry Highway during a 1 hour period. All speeds are mph. 45 39 55 42 52 39 46 61 54 33 50 47 29 57 60 Construct a frequency distribution, histogram, and frequency polygon for this information.

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

  11. Mean • 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.

  12. Example 2 Help Trooper Fife find the average speed along Mayberry Highway during that particular hour. 45 39 55 42 52 39 46 61 54 33 50 47 29 57 60

  13. Mean for Frequency Distributions

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

  15. Two Types of Mean • 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”)

  16. Median • 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.

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

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

  19. Example 4 Find the speed along Mayberry Highway that would represent the median speed. What speed(s) represent the mode, if any? 45 39 55 42 52 39 46 61 54 33 50 47 29 57 60

  20. Advantages and Disadvantages for Mean, Median, and Mode

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