1. Lecture 15 Sections 5.1 – 5.2 Wed, Sep 27, 2006 Measuring Center

2. Measuring the Center Often, we would like to have one number that that is “representative” of a population or sample. Often, we would like to have one number that that is “representative” of a population or sample. It seems reasonable to choose a number that is near the “center” of the distribution rather than in the left or right extremes. It seems reasonable to choose a number that is near the “center” of the distribution rather than in the left or right extremes. But there is no single “correct” way to do this. But there is no single “correct” way to do this.

3. Measuring the Center Mean – the simple average of a set of numbers. Mean – the simple average of a set of numbers. Median – the value that divides the set of numbers into a lower half and an upper half. Median – the value that divides the set of numbers into a lower half and an upper half. Mode – the most frequently occurring value in the set of numbers. Mode – the most frequently occurring value in the set of numbers.

4. Measuring the Center In a unimodal, symmetric distribution, these values will all be near the center. In a unimodal, symmetric distribution, these values will all be near the center. In skewed distributions, they will be spread out. In skewed distributions, they will be spread out.

5. The Mean Why is the average usually a good measure of the center? Why is the average usually a good measure of the center? If we have only two numbers, the average is half way between them. If we have only two numbers, the average is half way between them. What if we have more than two numbers? What if we have more than two numbers? The mean balances the “deviations” on the left with the “deviations” on the right. The mean balances the “deviations” on the left with the “deviations” on the right.

6. The Mean 12345678109

7. The Mean 12345678109 Average

8. The Mean 12345678109 Average -2 -5

9. The Mean 12345678109 Average -2 -5 +1 +2 +4

10. The Mean We use the letter x to denote a value from the sample or population. We use the letter x to denote a value from the sample or population. The symbol  means “add them all up.” The symbol  means “add them all up.” So, So,  x means add up all the values in the population or sample (depending on the context). Then the sample mean is Then the sample mean is

11. The Mean We denote the mean of a sample by the symbol  x, pronounced “x bar”. We denote the mean of a sample by the symbol  x, pronounced “x bar”. We denote the mean of a population by , pronounced “mu” (myoo). We denote the mean of a population by , pronounced “mu” (myoo). Therefore, Therefore,

12. TI-83 – The Mean Enter the data into a list, say L 1. Enter the data into a list, say L 1. Press STAT > CALC > 1-Var Stats. Press STAT > CALC > 1-Var Stats. Press ENTER. “1-Var-Stats” appears. Press ENTER. “1-Var-Stats” appears. Type L 1 and press ENTER. Type L 1 and press ENTER. A list of statistics appears. The first one is the mean. A list of statistics appears. The first one is the mean.

13. Examples Use the TI-83 to find the mean of the following data. Use the TI-83 to find the mean of the following data. 2.9462.3353.4181.890 2.7313.8551.3442.126 2.8812.5422.5043.367 1.9502.3922.4433.053

14. The Median 12345678109

15. The Median 12345678109 Median

16. The Median Median – The middle value, or the average of the middle two values, of a sample or population, when the values are arranged from smallest to largest. Median – The middle value, or the average of the middle two values, of a sample or population, when the values are arranged from smallest to largest. The median, by definition, is at the 50 th percentile. The median, by definition, is at the 50 th percentile. It separates the lower 50% of the sample from the upper 50%. It separates the lower 50% of the sample from the upper 50%.

17. The Median When n is odd, the median is the middle number, which is in position (n + 1)/2. When n is odd, the median is the middle number, which is in position (n + 1)/2. When n is even, the median is the average of the middle two numbers, which are in positions n/2 and n/2 + 1. When n is even, the median is the average of the middle two numbers, which are in positions n/2 and n/2 + 1.

18. The Median Suppose we surveyed a two groups of households and found the following numbers of children: Suppose we surveyed a two groups of households and found the following numbers of children: Group 1: 3, 2, 5, 2, 6. Group 1: 3, 2, 5, 2, 6. Group 2: 2, 3, 0, 2, 1, 0, 3, 0, 1, 4. Group 2: 2, 3, 0, 2, 1, 0, 3, 0, 1, 4. Find the median number of children in each group. Find the median number of children in each group.

19. TI-83 – The Median Follow the same procedure that was used to find the mean. Follow the same procedure that was used to find the mean. When the list of statistics appears, scroll down to the one labeled “Med.” It is the median. When the list of statistics appears, scroll down to the one labeled “Med.” It is the median.

20. TI-83 – The Median Use the TI-83 to find the medians of the samples Use the TI-83 to find the medians of the samples 3, 2, 5, 2, 6 3, 2, 5, 2, 6 2, 3, 0, 2, 1, 0, 3, 0, 1, 4 2, 3, 0, 2, 1, 0, 3, 0, 1, 4

21. The Median vs. The Mean If the data are strongly skewed, then the median is generally to give a more representative value. If the data are strongly skewed, then the median is generally to give a more representative value. If the data are not skewed, then the mean is usually preferred. If the data are not skewed, then the mean is usually preferred.

22. The Mode Mode – The value in the sample or population that occurs most frequently. Mode – The value in the sample or population that occurs most frequently. The mode is a good indicator of the distribution’s central peak, if it has one. The mode is a good indicator of the distribution’s central peak, if it has one.

23. Mode The problem is that many distributions do not have a peak or they have several peaks. The problem is that many distributions do not have a peak or they have several peaks. In other words, the mode does not necessarily exist or there may be several modes. In other words, the mode does not necessarily exist or there may be several modes.

24. Weighted Means Find the average number of children for each group. Find the average number of children for each group. Group 1: 3, 2, 5, 2, 6. Group 1: 3, 2, 5, 2, 6. Group 2: 2, 3, 0, 2, 1, 0, 3, 0, 1, 4. Group 2: 2, 3, 0, 2, 1, 0, 3, 0, 1, 4.

25. Weighted Means The averages are The averages are Group 1:  x 1 = 3.6. Group 1:  x 1 = 3.6. Group 1:  x 2 = 2.6. Group 1:  x 2 = 2.6. How could we combine the two averages to get the average of all the families together? How could we combine the two averages to get the average of all the families together?

26. Mean, Median, and Mode If a distribution is symmetric, then the mean, median, and mode are all the same and are all at the center of the distribution. If a distribution is symmetric, then the mean, median, and mode are all the same and are all at the center of the distribution.

27. Mean, Median, and Mode However, if the distribution is skewed, then the mean, median, and mode are all different. However, if the distribution is skewed, then the mean, median, and mode are all different.

28. Mean, Median, and Mode However, if the distribution is skewed, then the mean, median, and mode are all different. However, if the distribution is skewed, then the mean, median, and mode are all different. The mode is at the peak. The mode is at the peak. Mode

29. Mean, Median, and Mode However, if the distribution is skewed, then the mean, median, and mode are all different. However, if the distribution is skewed, then the mean, median, and mode are all different. The mode is at the peak. The mode is at the peak. The mean is shifted in the direction of skewing. The mean is shifted in the direction of skewing. Mode Mean

30. Mean, Median, and Mode However, if the distribution is skewed, then the mean, median, and mode are all different. However, if the distribution is skewed, then the mean, median, and mode are all different. The mode is at the peak. The mode is at the peak. The mean is shifted in the direction of skewing. The mean is shifted in the direction of skewing. The median is (typically) between the mode and the mean. The median is (typically) between the mode and the mean. ModeMedian Mean