Sometimes measures of central tendency alone do not give enough information for inferential statistics. Consistency or spread of data must be included before conclusions can be drawn. Example 1 on page 143 is a perfect example of how the mean, median and mode may not be enough to draw conclusions. George: mean = 185, median = 185, mode = 185 Danny: mean = 185, median = 185, mode = 185 Relying solely on central tendency draws an inaccurate conclusion about both bowlers. Measures of dispersion determine how data points differ from the average (mean). The difference between a single data point x, and the mean
x bar is called If all the deviations from the mean were added together, the total = 0 (by definition of mean). To use deviation, first square the difference before summing. Variance:
s s standard deviation =
Find the standard deviation of the following:
x = First find variance:
s
s = standard deviation = (Keep data on board for later) Alternate Formula:
s ^{2} ]
looks difficult, but is easier to use.n-1 n
Recall with grouped data; first find the midpoint x
s ^{2} ]n -1 n Using both measures of central tendency and measures of dispersion gives the best analysis of a collection of data. Combining both measures provides a look at what percentage lies within a specified number of standard deviations of the mean. Using EX 1 of typing errors, s = 1.8, x = 2, then 2 – 1.8 = .2 2 + 1.8 = 3.8 ___________|_________________
0 4/6 = 2/3 = .66 = 66% lie within one standard deviation of the mean.
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