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• The mean for quantitative data is obtained by dividing the sum of all values by the number of values in the data set. x x n The following are the ages of all eight employees of a small company: 53 32 61 27 39 44 49 57 Find the mean age of these employees. x 362 x 45.25 years n 8 Thus, the mean age of all eight employees of this company is 45.25 years, or 45 years and 3 months. The median is the value of the middle term in a data set that has been ranked in increasing order. The calculation of the median consists of the following two steps: 1.Sort/Arrange the data set in increasing order 2.Find the middle term in a data set with n values. The value of this term is the median. The following data give the weight lost (in pounds) by a sample of five members of a health club at the end of two months of membership: 10 5 19 8 3 Find the median. First, we rank the given data in increasing order as follows: 3 5 8 10 19 Therefore, the median is 3 5 8 10 19 The median weight loss for this sample of five members of this health club is 8 pounds. The median gives the “center” with half the data values to the left of the median and half to the right of the median. The advantage of using the median as a measure of central tendency is that it is less influenced by outliers & skewness. Consequently, the median is preferred over the mean as a measure of central tendency for data sets that contain outliers and/or skewness. The mode is the value that occurs with the highest frequency in a data set. • Range = Largest value – Smallest Value • The range, like the mean has the disadvantage of being influenced by outliers. • Its calculation is based on two values only: the largest and the smallest. • The standard deviation is the most used measure of dispersion. • The value of the standard deviation tells how closely the values of a data set are clustered around the mean. x 82 95 67 92 deviation 82 – 84 = -2 95 – 84 = +11 67 – 84 = -17 92 – 84 = +8 ∑(deviation) = 0 • standard deviation stdev = sqrt (sum squared deviations divided by n-1) Example :: sqrt[(4 + 121 + 289 + 64)/3] sqrt(478/3) = sqrt(159.3) = 12.62 • A numerical measure such as the mean, median, mode, range, variance, or standard deviation calculated for a population data set is called a population parameter, or simply a parameter. • A summary measure calculated for a sample data set is called a sample statistic, or simply a statistic. For a bell shaped distribution approximately 1. 68% of the observations lie within one standard deviation of the mean 2. 95% of the observations lie within two standard deviations of the mean 3. 99.7% of the observations lie within three standard deviations of the mean The age distribution of a sample of 5000 persons is bell-shaped with a mean of 40 years and a standard deviation of 12 years. Determine the approximate percentage of people who are 16 to 64 years old. Quartiles are three summary measures that divide a ranked data set into four equal parts. The second quartile is the same as the median of a data set. The first quartile is the value of the middle term among the observations in the lower half, and the third quartile is the value of the middle term among the observations in the upper half. Each of these portions contains 25% of the observations of a data set arranged in increasing order 25% 25% Q1 25% Q2 25% Calculating Interquartile Range The difference between the third and first quartiles gives the interquartile range; that is, IQR = Interquartile range = Q3 – Q1 The following are the ages of nine employees of an insurance company: 47 28 39 51 33 37 59 24 33 a)Find the values of the three quartiles. Where does the age of 28 fall in relation to the ages of the employees? b)Find the interquartile range. The following data are the incomes (in thousands of dollars) for a sample of 12 households. 35 29 44 72 34 64 41 50 54 104 39 58 Construct a box-and-whisker plot for these data. Step 1. 29 34 35 39 41 44 50 54 58 64 72 104 Median = (44 + 50) / 2 = 47 Q1 = (35 + 39) / 2 = 37 Q3 = (58 + 64) / 2 = 61 IQR = Q3 – Q1 = 61 – 37 = 24 Step 2. 1.5 x IQR = 1.5 x 24 = 36 Lower inner fence = Q1 – 36 = 37 – 36 = 1 Upper inner fence = Q3 + 36 = 61 + 36 = 97 Step 3. Smallest value within the two inner fences = 29 Largest value within the two inner fences = 72 First quartile Third quartile Median 25 35 45 55 65 Income 75 85 95 105 Smallest value within the two inner fences Third First quartile quartile Median An outlier Largest value within two inner fences 25 35 45 55 65 Income 75 85 95 105 What Can Go Wrong? • Do a reality check— don’t let technology do your thinking for you. • Don’t forget to sort the values before finding the median … quartiles. • Don’t compute numerical summaries of a categorical variable. • Watch out for multiple peaks—multiple peaks might indicate multiple groups in your data. What Can Go Wrong? • Be aware of slightly different methods— different statistics packages and calculators may give you different answers for the same data. • Beware of outliers. • Make a picture (make a picture, make a picture). • Be careful when comparing groups that have very different spreads. So What Do We Know? • We describe distributions in terms of L.O.S.S. • For symmetric distributions, it’s safe to use the mean and standard deviation; for skewed distributions, it’s better to use the median and interquartile range. • Always make a picture—don’t make judgments about which measures of center and spread to use by just looking at the data.