Measures of Dispersion Section 4.3

The case of Fred and Barney at the bowling alley Fred and Barney are at the bowling alley and they want to know who’s the better bowler. They bowl six games and here are the results: Fred Barney

Find the average, Find the median, Find the mode!!! After the games it’s time for Fred and Barney to do the math. We compute Fred’s mean and we see it is 185. Barney’s mean is computed and it is also 185. We look at Fred’s median it’s 185. Barney’s median is also 185. The mode for Fred is 185 and the mode for Barney is 185. Make sure you can do these calculations.

Mr. Consistency vs. Mr. Loose Cannon If we look at the scores we notice that Barney’s scores are very consistent. They do not vary much around his average. Fred however has wildly varying scores. His last game was 250, so maybe he’s Mr. Clutch. Statistically speaking we would like to measure this variation about the mean. What we need to do is to calculate deviations from the mean, sample variance, and sample standard deviation.

Deviation from the mean First let’s calculate the deviation from the mean for each score. The formula is. For Fred the avg. of his scores is 185. d 1 = 185 – 185 = 0 d 2 = 135 – 185 = -50 d 3 = 200 – 185 = 15 d 4 = 185 – 185 = 0 d 5 = 155 – 185 = -30 d 6 = 250 – 185 = 65 The sum of the deviations is zero! This is always true. What we need to do is get rid of those pesky minus signs.

The sample variance The measurement we need is called the sample variance. What we do is we square each deviation and then sum them up and divide by one less the number of data points. The formula is give as:

The Sample variance for Fred Lets calculate the sample variance for Fred.

The standard deviation. Since we want an understanding of how the data is dispersed about the mean, then the statistic that measures this must be of the same units as the mean. Unfortunately the sample variance is the square of these units. So what we should do is take the square root of the sample variance. This is called the sample standard deviation. Sample Standard Deviation = sample variance

The standard deviation for Fred We can now calculate Fred’s sample standard deviation. You should calculate Barney’s sample variance and sample standard deviation. They are

One Standard deviation from the mean Sometimes it is useful to compute the percentage of the data that is one standard deviation from the mean. What you need to do is first compute the mean of the sample. Then compute the standard deviation of the sample. Next, you want to construct the interval that is one standard deviation from the mean. The left end point is. The right end point is. Find the number of data points that fall into this range and then divide by the total number of data points. When this number is close to 68%, this is indicative of a normal distribution.