Measures of Central Tendency What does a typical car cost? Who is the best quarterback in Pro Football? What is the most popular soft drink? These questions can be answered using “average values”. Car prices based on average prices, quarterbacks compared by their averages: number of passes/completions/yards. Soft drinks by average sales, taste tests. Instead of listing data points for a large distribution of values, we can summarize the data by selecting a representative value, calling it the “average”. Three values describe “average”: mean, median and mode. All three values describe the “center” of a distribution of numbers. Collectively, they are called measures of central tendency. Mean: Add the values of data, then divide by the number of data points. Sample mean: x Example 1: Find the mean, x, test score of the sample x1= 85 x2= 69 x3= 75 x4= 62 x5 = 80 x6 = 94
85 + 69 + 75 + 62 + 80 + 94 =
465 = 77.5 Shorthand for Summation:
Sx = x = 77.5 What if data is grouped? Then find the midpoint of each interval and multiply by the
frequency, f.
Sfx
Sometimes the mean can be misleading.
Example 2: Suppose our data is home prices in a certain neighborhood: $110,000 $90,000 $120,000 $100,000 $90,000 $450,000 x = 960,000/6 = $160,000 is not the typical cost of a house in this neighborhood even though it is the mean. One of the data points is an outlier (extreme value). In this case, the mean is not a good indication for average value. Median: the “Middle value” of a distribution of numbers.
Using EX 2, place the data points in ascending order numerically. $90,000 $90,000 $100,000 $110,000 $120,000 $450,000 The median value is the one in the middle. If there is an even number of data points, take the two middle values, add them together and divide by two.
$100,000 + $110,000 = $105,000 $105,000 is the median value, another method of determining average. Mode: the most frequent number in a collection of data. Using the last example, the mode is $90,000 because it occurs more often than any other value. Sometimes a sample may not have a mode, or it may be bi-modal, having two values with the same high frequency. When summarizing distribution of values, it is best to list all three measures of central tendency to avoid misleading or confusing situations where the word average is used. Information can be misused and the public manipulated by statistical methods and terms.
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