Mean Standard Deviation Grouped Data
- Page ID
- 64199
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Calculating the Mean from a Frequency Distribution Since calculating the mean and standard deviation is tedious, we can save some of this work when we have a frequency distribution. Suppose we were interested in how many siblings are in statistics students' families. We come up with a frequency distribution table below.
Notice that since there are 29 respondents, calculating the mean would be very tedious. Instead, we see that there are five ones, 12 twos, 8 threes, 3 fours, and 1 seven. Hence the total count of siblings is 1(5) + 2(12) + 3(8) + 4(3) + 7(1) = 72 Now divide by the number of respondents to get the mean. \( m = \frac{72}{29} = 25 \)
Extending the Frequency Distribution Table Just as with the mean formula, there is an easier way to compute the standard deviation given a frequency distribution table. We extend the table as follows:
Next we calculate (Sxf)2 (72)2 = 43.24 Now finally apply the formula \(s = \sqrt{\frac{SS_x}{n - 1}} \) to get \(s = \sqrt{\frac{43.24}{29 - 1}} = 1.24 \) Click here for an applet that finds statistics from grouped data. Weighted Averages Sometimes instead of the simple mean, we want to weight certain outcomes higher then others. For example, for your statistics class, the following percentages are given Homework = 150 Midterm = 450 Project = 100 Final = 300 Suppose that you received an 88% on your homework, a 97% on your midterms, a 98% on your project and an 78% on your final. What is your average for you class? To compute the weighted average, we use the formula \( \text{Weighted Average} = \frac{S_{xw}}{S_{w}} \) We have Sxw = .88(150) + .97(450) + .98(100) + .78(300) = 900.5 and Sw = 150 + 450 + 100 + 300 = 1000 Now divide to get your weighted average \( \frac{900.5}{1000} = 0.9005 \) You squeaked by with an "A". Back to the Descriptive Statistics Home Page
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