1. Measures of Central Tendency Mean, Median, Mode, Range, Outliers, and Measure of Center

2. What’s the Objective? As we review the concepts of measures of central tendency, we will explore the idea of outliers and measures that best represent the data – both numerical and non-numerical. This measure that best represents data may be called the “measure of center.”

3. Vocabulary Mean: Median: Mode: Range: Outlier: Measure of Center : Sum of data values Number of data values added When arranged in descending or ascending order, the value in the middle The value listed more than any other; there may be more than one mode. There may not be a mode. The difference in the greatest and least values. Any value that is much greater or much less than the other values. There may not be an outlier. The measure that BEST represents the data set. This may be the mean, median or mode!

4. The Lesson In a class Readathon, six students entered the number of pages they had read during class. The pages entered were: 50, 42, 45, 48, 59, and 50. 50 + 42 + 45 + 48 + 59 + 50 6 = 294 6 = 49 This represents the mean (average) number of pages read. The mode is 50. It is listed more than any other. The range?17 Since there are an even number of values, we will have two numbers in the middle. Numbers must be listed in order to find the median!!!!!!!!!!!! Ordered: 42, 45, 48, 50, 50, 59 The median is 49. We have two numbers in the “middle” so we find the middle of those.

5. Let’s use another data set to work with these measures. Test Scores: 100, 90, 75, 80, 85, 90, 90, 30. Write them in order as you find the mean. It will save time. 80 = mean Once again we will have two numbers in the middle. This is because we have an even number of data elements. 85 and 90 are in the middle. You can find the average of these two, or you can just ask yourself, “What’s in the middle of these two numbers?” The median is 87.5

6. Test Scores: 100, 90, 75, 80, 85, 90, 90, 30 What is the mode of the scores? The mode is 90. It is listed more than the others. What is the range? The range is 70……… (100 – 30)

7. What is an outlier? 100, 90, 75, 80, 85, 90, 90, 30 Is there a score that is much lower (or higher) than all the other numbers? This is an outlier. In this case, 30 is the outlier. Did it affect the mean???????? The mean with the outlier was: 640 8 = 80 What would the mean be without the outlier? 640 - 30 7 ~ 87 The effect of the outlier was to lower the average.

8. What is the Measure of Center? Well, it depends. It may be the mean. It may be the median. It may also be the mode. Let’s look at the previous two sets of data to discuss these possibilities.

9. In the Readathon data, were there any values that were outliers? Ordered: 42, 45, 48, 50, 50, 59 Not really. All of these values were reasonably close to each other. The mean was 49. The median was also 49. In this case, either the mean or the median could be the best measure of center. You would usually choose the mean if there are no outliers.

10. The test scores, however, were not all closely centered. Let’s use a line plot to look at this uneven distribution. 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 XX X XXXXXX X X Most of the scores are clustered in this area. The data is skewed to the left. This outlier will change what we call our measure of center. The mean, with the outlier, was an 80. The median was 87.5. Which of these two measures BEST represents MOST of the scores in the set? The effect the outlier had on the mean, indicates that we would use the median as the measure of center. The mean was not the “best” representation of all the scores.

11. Let’s look at this chart of a company’s hourly wages for its employees. In this case, the median would best represent the majority of wages. The outlier of $17/hour raised the average wage to a value much higher that most wages. The median is the measure of center. The mean is $7.50. The median is $6.10. Which measure BEST represents all of the data? Why? $7.50 mean Median $6.10

12. Determining the Measure of Center Since 39 miles is an outlier, the median would be the best measure to represent the survey results. The median would be the measure of center. We could say this was skewed to the right. This means the cluster is on the left. In a survey of miles from home to school, the results were: 5, 6, 6, 7, 7, 7,10, 12, 39. Is there an outlier? This should give us an indication of our best measure to represent the data. Let’s check it. (5 + 6 + 6 + 7 + 7 + 7 + 10 + 12 + 39) ÷ 9 = mean 99 ÷ 9 = 11 miles mean What is the median? We have 9 data elements, so the middle is …… The median is 7. Which measure, the mean or the median would be the measure of center? Which one BEST represents the data? 5 10 15 20 25 30 35 40 X XXXX XXXXXX XXX The mode is 7.

13. In a quality check of how many potatoes are in a 3 lb. bag, inspectors found: 12, 13, 14,13, 11, 12, 13, 14, 15, 8, 16, and 9 potatoes in the bags. There are no outliers. The best measure for this quality check is the mean, so measure of center for this data set would be the mean. 0 2 4 6 8 10 12 14 16 18 XX X X X X X X X The distribution of this data would be symmetric. It is not skewed (clustered) on one side or the other. XX X

14. Some data sets do not contain numbers. For example, the circle graph shows the result of a survey to find people’s favorite color. When it does not contain numbers the only way to describe the data set is with the mode. You cannot find a mean or a median for a set of colors. The mode for this data set is blue. Most people in this survey chose blue as their favorite color. Group Discussion

15. Choosing the Best Measure Thriller Comedy Action/Adventure Thriller Action/Adv Comedy Romance What measure – mean, median, or mode – would be best to represent this survey, the measure of center? You would use the mode. Non-numerical data is represented by the mode. It is all you have! In a survey of favorite types of movies, the responses were:

16. Favorite Movies The data would not be numerical. Science fiction, fantasy, western, suspense, western, thriller The mode best represents non-numerical data. These are the result of a survey about favorite flavors of candy: grape, grape, banana, nectarine, strawberry, strawberry, strawberry, orange, watermelon Strawberry is the non-numerical mode.

17. Did we meet our objective? What was the objective???