Lecture 15 Sections 5.1 – 5.2 Fri, Oct 1, 2004 Measuring Center Lecture 15 Sections 5.1 – 5.2 Fri, Oct 1, 2004

Measuring the Center 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. There are several standard ways to do this.

The Mean Mean – the simple average of a set of numbers.

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

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

TI-83 – The Mean Enter the data into a list, say L1. Press STAT > CALC > 1-Var Stats. Press ENTER. “1-Var-Stats” appears. Type L1 and press ENTER. A list of statistics appears. The first one is the mean. See p. 269 for more details.

Let’s Do It! Find the mean of the age data in Data Set 1, p. 268. Think About It, p. 270 – Is the Mean Always the Center? Be Careful! The mean can be at any percentile, although it is usually near the 50th percentile. Let’s do it! 5.1, p. 270 – A Mean is not Always Representative. Let’s do it! 5.2, p. 270 – Combining Means.

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. The median, by definition, is at the 50th percentile.

The Median When n is odd, the median is the middle number, which is in position (n + 1)/2. Example: Find the median of 10, 12, 7, 9, 14. When n is even, the median is the average of the middle two numbers, which are in positions n/2 and n/2 + 1. Example: Find the median of 10, 12, 7, 9, 14, 11.

TI-83 – The Median Enter the data into a list, say L1. Press STAT > CALC > 1-Var Stats. Press ENTER. Type L1 and press ENTER. A list of statistics appears. The one labeled “Med” is the median. See p. 272 for more details.

TI-83 – The Median An alternate method is to sort the data. Enter the data into a list, say L1. Press LIST > OPS > SortA. Press ENTER. Type L1 and press ENTER. The word “Done” appears. The list is now sorted from smallest to largest. Type L1 and press ENTER to see the list, or view it in the Edit mode.

The Median vs. The Mean In the last example, change 14 to 14000 and recompute the median. How did the change affect the median? How did the change affect the mean? Which is a better measure of the center of this sample?

Let’s Do It! Let’s do it! 5.3, p. 272 – Median Number of Children per Household.

The Mode 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.

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

Example Example 5.3, p. 274 – Different Measures Can Give Different Impressions. Which measure (mean, median, or mode) is most representative of the sample?

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.

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

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

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

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

Let’s Do It! Let’s do it! 5.5, p. 276 – Attend Graduate School? Is the sample skewed? If so, in which direction is it skewed?