Lecture 15 Sections 5.1 – 5.2 Tue, Feb 14, 2006 Measuring Center Lecture 15 Sections 5.1 – 5.2 Tue, Feb 14, 2006

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. But there is no single “correct” way to do this.

Measuring the Center 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. Mode – the most frequently occurring value in the set of numbers.

Measuring the Center In a unimodal, symmetric distribution, these values will all be near the center. In other-shaped distributions, they may be spread out.

The Mean 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

The Mean 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). 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. 301 for more details.

Examples Use the TI-83 to find the mean of the data in Example 5.1(a), p. 301.

Weighted Means Continuing the previous example, suppose we surveyed another group of households and found the following number of children: 3, 2, 5, 2, 6. Find the average of this group by itself. Combine the two averages into one average for all 15 households.

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. It separates the lower 50% of the sample from the upper 50%.

The Median When n is odd, the median is the middle number, which is in position (n + 1)/2. Find the median of 3, 2, 5, 2, 6. When n is even, the median is the average of the middle two numbers, which are in positions n/2 and n/2 + 1. Find the median of 2, 3, 0, 2, 1, 0, 3, 0, 1, 4.

The Median Alternately, we could calculate (n + 1)/2 in all cases. If it is a whole number, then use that position. If it is halfway between two whole numbers, take use both positions and take the average.

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

The Median vs. The Mean In the last example, change 4 to 4000 and recompute the mean and 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?

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 or there may be several modes.

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! 5.6, p. 309 – A Different Distribution. Do the same for the distribution 4 1/2 x