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Means & Medians

Means & Medians. Chapter 4. Parameter -. Suppose we want to know the MEAN length of fish in Lake Lewisville. Fixed value about a population Typical unknown. Statistic -. Value calculated from a sample. Measures of Central Tendency. Median - the middle of the data; 50 th percentile

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Means & Medians

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  1. Means & Medians Chapter 4

  2. Parameter - Suppose we want to know the MEAN length of fish in Lake Lewisville . . . • Fixed value about a population • Typical unknown

  3. Statistic - • Value calculated from a sample

  4. Measures of Central Tendency • Median - the middle of the data; 50th percentile • Observations must be in numerical order • Is the middle single value if n is odd • The average of the middle two values if n is even NOTE:n denotes the sample size

  5. Measures of Central Tendency parameter • Mean - the arithmetic average • Use m to represent a population mean • Use x to represent a sample mean statistic • Formula: S is the capital Greek letter sigma – it means to sum the values that follow

  6. Measures of Central Tendency • Mode – the observation that occurs the most often • Can be more than one mode • If all values occur only once – there is no mode • Not used as often as mean & median

  7. Suppose we are interested in the number of lollipops that are bought at a certain store. A sample of 5 customers buys the following number of lollipops. Find the median. The numbers are in order & n is odd – so find the middle observation. The median is 4 lollipops! 2 3 4 8 12

  8. Suppose we have sample of 6 customers that buy the following number of lollipops. The median is … The numbers are in order & n is even – so find the middle two observations. The median is 5 lollipops! Now, average these two values. 5 2 3 4 6 8 12

  9. Suppose we have sample of 6 customers that buy the following number of lollipops. Find the mean. To find the mean number of lollipops add the observations and divide by n. 2 3 4 6 8 12

  10. Using the calculator . . .

  11. What would happen to the median & mean if the 12 lollipops were 20? 5 The median is . . . 7.17 The mean is . . . What happened? 2 3 4 6 8 20

  12. What would happen to the median & mean if the 20 lollipops were 50? 5 The median is . . . 12.17 The mean is . . . What happened? 2 3 4 6 8 50

  13. Resistant - • Statistics that are not affected by outliers • Is the median resistant? YES NO • Is the mean resistant?

  14. Look at the following data set. Find the mean. 22 23 24 25 25 26 29 30 Now find how each observation deviates from the mean. What is the sum of the deviations from the mean? Will this sum always equal zero? This is the deviation from the mean. YES 0

  15. Look at the following data set. Find the mean & median. Mean = Median = 27 27 Create a histogram with the data. (use x-scale of 2) Then find the mean and median. Look at the placement of the mean and median in this symmetrical distribution. 21 23 23 24 25 25 26 26 26 27 27 27 27 28 30 30 30 31 32 32

  16. Look at the following data set. Find the mean & median. Mean = Median = 28.176 25 Create a histogram with the data. (use x-scale of 8)Then find the mean and median. Look at the placement of the mean and median in this right skewed distribution. 22 29 28 22 24 25 28 21 25 23 24 23 26 36 38 62 23

  17. Look at the following data set. Find the mean & median. Mean = Median = 54.588 58 Create a histogram with the data. Then find the mean and median. Look at the placement of the mean and median in this skewed left distribution. 21 46 54 47 53 60 55 55 60 56 58 58 58 58 62 63 64

  18. Recap: • In a symmetrical distribution, the mean and median are equal. • In a skewed distribution, the mean is pulled in the direction of the skewness. • In a symmetrical distribution, you should report the mean! • In a skewed distribution, the median should be reported as the measure of center!

  19. Example calculations • During a two week period 10 houses were sold in Fancytown. The “average” or mean price for this sample of 10 houses in Fancytown is $295,000

  20. Outlier • During a two week period 10 houses were sold in Lowtown. The “average” or mean price for this sample of 10 houses in Lowtown is $295,000

  21. Outlier • Looking at the dotplots of the samples for Fancytown and Lowtown we can see that the mean, $295,000 appears to accurately represent the “center” of the data for Fancytown, but it is not representative of the Lowtown data. • Clearly, the mean can be greatly affected by the presence of even a single outlier.

  22. In the previous example of the house prices in the sample of 10 houses from Lowtown, the mean was affected very strongly by the one house with the extremely high price. • The other 9 houses had selling prices around $100,000. • This illustrates that the mean can be very sensitive to a few extreme values. SOOOO……

  23. Describing the Center of a Data Set with the median The sample median is obtained by first ordering the n observations from smallest to largest (with any repeated values included, so that every sample observation appears in the ordered list). Then

  24. Example of Median Calculation Consider the Fancytown data. First, we put the data in numerical increasing order to get 231,000 285,000 287,000 294,000 297,000 299,000 312,000 313,000 315,000 317,000 Since there are 10 (even) data values, the median is the mean of the two values in the middle.

  25. Consider the Lowtown data. We put the data in numerical increasing order to get 93,000 95,000 97,000 99,000 100,000 110,000 113,000 121,000 122,000 2,000,000 Since there are 10 (even) data values, the median is the mean of the two values in the middle.

  26. mean vs median

  27. Typically, • when a distribution is skewed positively, the mean is larger than the median, • when a distribution is skewed negatively, the mean is smaller then the median, and • when a distribution is symmetric, the mean and the median are equal.

  28. The Trimmed Mean • A trimmed mean is computed by first ordering the data values from smallest to largest, deleting a selected number of values from each end of the ordered list, and finally computing the mean of the remaining values. • The trimming percentage is the percentage of values deleted from each end of the ordered list.

  29. Example of Trimmed Mean

  30. Example of Trimmed Mean

  31. Trimmed mean: Purpose is to remove outliers from a data set To calculate a trimmed mean: • Multiply the % to trim by n • Truncate that many observations from BOTH ends of the distribution (when listed in order) • Calculate the mean with the shortened data set

  32. Find a 10% trimmed mean with the following data. 12 14 19 20 22 24 25 26 26 35 10%(10) = 1 So remove one observation from each side!

  33. Example: The Highway Loss Data Institute publishes data on repair costs resulting from a 5-mph crash test of a car moving forward into a flat barrier. The following table gives data for 10 midsize luxury cars: ModelRepair Cost in dollars Audi A6 0 BMW 328i 0 Cadillac Catera 900 Jaguar X 1254 Lexus ES300 234 Lexus IS 300 979 Mercedes C320 707 Saab 9-5 670 Volvo S60 769 Volvo S80 4194 Compute the mean and median values. Why are these values so different? Which of the mean and median do you think is more representative of the data set? Why?

  34. Test day: The average for Mrs. Goins’ 4th period Algebra test was 88% for those 24 students. The average for 6th period was 85% for those 20 students. What was the average for the combined classes?

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