Two Data Sets Stock A Stock B Closing prices for two stocks were recorded on ten successive Fridays. Calculate the mean, median and mode for each. 56 33 56 42 57 48 58 52 61 57 63 67 67 77 67 82 67 90 Stock A Stock B Use this example to review the measures of central tendency. Both sets of data have the same mean, the same median and the same mode. Students clearly see that the data sets are vastly different though. A good lead in for measures of variation. Mean = 61.5 Median = 62 Mode = 67 Mean = 61.5 Median = 62 Mode = 67

Measures of Variation Range = Maximum value – Minimum value Range for A = 67 – 56 = $11 Range for B = 90 – 33 = $57 The range is easy to compute but only uses two numbers from a data set. Explain that if only one number changes, the range can be vastly changed. If a $56 stock dropped to $12 the range would change. Emphasize the need for different symbols for population parameters and sample statistics.

Measures of Variation To learn to calculate measures of variation that use each and every value in the data set, you first want to know about deviations. The deviation for each value x is the difference between the value of x and the mean of the data set. Explain that if only one number changes, the range can be vastly changed. If a $56 stock dropped to $12 the range would change. Emphasize the need for different symbols for population parameters and sample statistics. In a population, the deviation for each value x is: In a sample, the deviation for each value x is:

Deviations Stock A Deviation 56 57 58 61 63 67 – 5.5 – 4.5 – 3.5 – 0.5 1.5 5.5 56 – 61.5 56 – 61.5 57 – 61.5 58 – 61.5 Make sure students know that the sum of the deviations from the mean is always 0. The sum of the deviations is always zero.

Population Variance Population Variance: The sum of the squares of the deviations, divided by N. x 56 57 58 61 63 67 – 5.5 – 4.5 – 3.5 – 0.5 1.5 5.5 30.25 20.25 12.25 0.25 2.25 188.50 Each deviation is squared to eliminate the negative sign. Students should know how to calculate these formulas for small data sets. For larger data sets they will use calculators or computer software programs. Sum of squares

Population Standard Deviation Population Standard Deviation: The square root of the population variance. The variance is expressed in “square units” which are meaningless. Using a standard deviation returns the data to its original units. The population standard deviation is $4.34.

Sample Variance and Standard Deviation To calculate a sample variance divide the sum of squares by n – 1. The sample standard deviation, s, is found by taking the square root of the sample variance. Samples only contain a portion of the population. This portion is likely to not contain extreme values. Dividing by n-1 gives a higher value than dividing by n and so increases the standard deviation.

Summary Range = Maximum value – Minimum value Population Variance Population Standard Deviation Sample Variance A summary of the formulas Sample Standard Deviation

Empirical Rule (68-95-99.7%) Data with symmetric bell-shaped distribution have the following characteristics. 13.5% 13.5% 2.35% 2.35% –4 –3 –2 –1 1 2 3 4 This type of distribution will be studied in much more detail in later chapters. About 68% of the data lies within 1 standard deviation of the mean About 95% of the data lies within 2 standard deviations of the mean About 99.7% of the data lies within 3 standard deviations of the mean

Using the Empirical Rule The mean value of homes on a street is $125 thousand with a standard deviation of $5 thousand. The data set has a bell shaped distribution. Estimate the percent of homes between $120 and $135 thousand. 125 130 135 120 140 145 115 110 105 $120 thousand is 1 standard deviation below the mean and $135 thousand is 2 standard deviations above the mean. 68% + 13.5% = 81.5% So, 81.5% have a value between $120 and $135 thousand.

Chebychev’s Theorem For any distribution regardless of shape the portion of data lying within k standard deviations (k > 1) of the mean is at least 1 – 1/k2. For k = 2, at least 1 – 1/4 = 3/4 or 75% of the data lie within 2 standard deviation of the mean. The empirical rule is valid for symmetric, mound -shaped distributions. Chebychev’s theorem applies to all distributions. Emphasize the phrase “at least” in the statement of the theorem. For k = 3, at least 1 – 1/9 = 8/9 = 88.9% of the data lie within 3 standard deviation of the mean.

Chebychev’s Theorem The mean time in a women’s 400-meter dash is 52.4 seconds with a standard deviation of 2.2 sec. Apply Chebychev’s theorem for k = 2. Mark a number line in standard deviation units. 2 standard deviations A Apply Chebychev’s theorem for k = 3. At least 1-19 = 8/9 or the data values will fall within 3 standard deviation of the mean no matter what the distribution. At least 8/9 of the times in this race will fall between 45.8 and 59 sec. 45.8 48 50.2 52.4 54.6 56.8 59 At least 75% of the women’s 400-meter dash times will fall between 48 and 56.8 seconds.