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Average, Median, and Standard Deviation

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## Average, Median, and Standard Deviation

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**Center and Spread**• The average and the median are measures of the center. • The standard deviation is a measure of the spread.**Center and Spread**The centers of the two histograms are the same, but the histogram on the right is more spread out.**Average, Mean, Median**• The average of a list of numbers equals their sum, divided by how many there are. • The average is sometimes called the (arithmetic) mean.**The Median**• The median of a list is the value with half the entries to the left, and half the entries to the right. • Find the median of 9, 1, 2, 2, 0: Order the list: 0, 1, 2, 2, 9 • Find the median of 7, 2, 1, 5: Order the list: 1, 2, 5, 7. The average of 2 and 5 is 3.5**Average vs. Median Income**• For persons age 25 and over in the U.S. would the average or the median be higher for income? For years of schooling completed?**The Tail of a Histogram**Average is bigger than the median Average is about the same as the median Average is smaller than the median**Standard Deviation**The standard deviation tries to quantify the spread.**The root-mean-square**• The average of the list 0, 5, -8, 7, -3 is 0.2. • The positive numbers ‘wipe out’ the negative ones. • We could neglect the signs. The average of 0, 5, 8, 7, 3 is 4.6 • Instead, we use the root-mean-square.**The root-mean-square**• Square all the entries. This will get rid of all the signs. • Take the mean (average) of the squares. • Take the square root of the mean.**Example**• Find the root-mean-square of 0, 5, -8, 7, -3. • Square: 0, 25, 64, 49, 9. • Mean: (0 + 25 + 64 + 49 + 9) : 5 = 29.4 • Root:**Computing the Standard Deviation**Find the standard deviation of 20, 10, 15, 15. • Find the average: 15. • Find the deviations from the average: 5, -5, 0, 0. • Find the root-mean-average:**Which list has the larger SD?**• 50, 40, 60, 30, 70, 25, 75 • 50, 40, 60, 30, 70, 25, 75, 50, 50, 50**68 – 95 - 99**• In many cases, roughly 68 percent of the entries on a list are within 1 SD of average; roughly 95% within 2 SDs; 99 percent are within 3 SDs of the average.**Example**• Average height for boys age 11 is 146 cm and the SD was 8 cm. • How much below average is a boy 170 cm tall? 148 cm tall? • How tall is a boy 1.5 SD below average? • Four boys are 150 cm, 130 cm, 165 cm, 140 cm. Who is unusually short, about average, unusually tall?