Lesson 3

1. Lesson 3 Measures of Central Tendency Lesson 3 Central Tendency

2. Central Tendency • What does Central Tendency mean? • Central tendency is one way in which we organize, summarize, and describe our data (and thereby the population and/or sample). • It tells us where the “middle” is. Lesson 3 Central Tendency

3. Central Tendency There are three ways to measure centrality (the “middle”): • The mode • The median • The mean Lesson 3 Central Tendency

4. Mode Frequency Distribution Table Finding the mode is easy! It is the score that occurs the most frequently. It is not the frequency of that score! Lesson 3 Central Tendency

5. median • The median is just a little bit harder to find. • Use a frequency distribution table or arrange the data from the smallest value to the largest value. Don’t leave out any values. If a 5 appears ten times, write it down ten times! • Find the middle of the distribution. Lesson 3 Central Tendency

6. median • Let’s find the median for this data set. Lesson 3 Central Tendency

7. Median • What is the median of this data set? Lesson 3 Central Tendency

8. Median • So what have we learned about the median? • Arrange the values from smallest to largest. • If the dataset has an odd number of values, the median will be one of the values from the dataset (see the first example). • If the dataset has an even number of values, the median will be halfway between the two middle values (see the second example). Lesson 3 Central Tendency

9. Mean • The mean (or average) is your first big statistical challenge!(population) (sample) In English—add up all the values for the variable X and then divide by the number of items you added. Lesson 3 Central Tendency

10. Mean Let’s find the mean for this data set. Lesson 3 Central Tendency

11. Mean Remember, the formula simply means you are add up all the values for the variable X and then divide by the number of items you added. Now let’s use our formula. Step 1. Add up all the values Step 2. Divide by the number of values added Lesson 3 Central Tendency

12. Mean When you want to find the mean from a frequency distribution table, you first multiply each X (score) by its associated f (frequency) to find fX. Adding up the fX values gives you the sum of X. Lesson 3 Central Tendency

13. Mean • See, adding up the fX column gives you then the SX. • By dividing SX by N, you can find the mean. Lesson 3 Central Tendency

14. Population vs. Sample Reminder: • m is the mean of a population. It is the population parameter. • X is the mean of a sample. It is the sample statistic. Lesson 3 Central Tendency

15. Relative Locations—Symmetric Distributions Mean, Median, & Mode Lesson 3 Central Tendency

16. Relative Locations—Positively Skewed Distributions Mode & Median Mean Lesson 3 Central Tendency

17. Relative Locations—Negatively skewed distributions Mode & Median Mean Lesson 3 Central Tendency

18. From bars to curves If the data intervals get smaller... Lesson 3 Central Tendency

19. From bars to curves . . . and smaller. . . Lesson 3 Central Tendency

20. From bars to curves . . .until you have a smooth curve like this. Lesson 3 Central Tendency

21. Relative Positions, again When the distribution is symmetrical-- Mode Median Mean Lesson 3 Central Tendency

22. Relative Positions, again If the distribution is positively skewed-- Low Scores High Scores Lesson 3 Central Tendency

23. Relative Positions, again If the distribution is negatively skewed-- Low Scores High Scores Lesson 3 Central Tendency

24. Relative Positions • You can tell a lot about a distribution just by looking at its three measures of central tendency. • If the mean, median, and mode all have about equal values, then it means the distribution is fairly symmetrical. • If the mode is a lower value than the median and the median is a lower value than the mean, then the distribution is positively skewed. • If the mode is a higher value than the median and the median is a higher value than the mean, then the distribution is negatively skewed. Lesson 3 Central Tendency

25. When to use what Lesson 3 Central Tendency

26. When to use what Lesson 3 Central Tendency

27. When to use what Lesson 3 Central Tendency

28. When to use what Lesson 3 Central Tendency

29. what do we know now? • When our data are nominal, only the mode can be used. • When are data are ordinal, we can maybe (but not always) find the median. But we can always find the mode. Lesson 3 Central Tendency

30. What do we know now? • When our data are interval or ratio, we can find the mean, the median, and the mode, but the mean may not always be representative. • When we have interval or ratio data and the distribution is skewed, the mean is a poor measure of central tendency. Lesson 3 Central Tendency