Section 3.1

Measures of Central Tendency: Mode, Median, and Mean

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Measures of Central Tendency: Mode, Median, and Mean

Usually, one number is used to describe the entire sample or population – the average.

three of the major ways to measure center of data:

1.Mode

2.Median

3.Mean

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Mode

- Data value that occurs the most- Not every data set has a mode

(Ex: professor assigns equal # of A’s, B’s, C’s, D’s, F’s)- Mode is not stable- Think tallest bar on a histogram- Most in a class (ex: bimodal means 2 modes)

- Relevant in cases like most frequently requested shoe size

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- Order data from smallest to largest- 50% of the data below and 50% above the median

Ex:

Data on price per ounce in cents of chips:

19 19 27 28 18 35

a) Mode?

b) Median?

c) Average?

d) What if you add 80 to the data set?

Median

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If we take out 35 from the data.

Median = 19

e) Is $10.45 reasonable to serve an ounce of chips to 55 people?

Yes, the median price of the chips is 19 cents per ounce.

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NOTE #1: The median uses the position - extreme values usually does not change it much.

Ex: the median is often used as the average for house prices.

NOTE #2: Extreme values inflate or deflate the average (mean)

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Mean

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A resistant measure is one that is not influenced by extremely high or low data values.

***The mean is not a resistant measure of center

***The median is more resistant measure of center

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Trimmed Mean

***More resistant than the regular mean

-- trim the lowest 5% of the data and highest 5% of the data (works the same for a 10% trimmed mean)

Procedure:

1.Order data

2.Multiply 5% by n and round to the nearest integer

3.that value is how many data points you trim from each end

4.Take the average of the remaining values

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Measures of Central Tendency: Mode, Median, and Mean

Symmetrical data: mean, median, and mode are the same or almost the same.

Left-Skewed data: mean < median and

median < mode

Right-Skewed data: mean > median and

Median > mode

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Relationship: Mode, Median, and Mean

Figure, shows the general relationships among the mean, median, and mode for different types of distributions.

(a) Mound-shaped symmetrical (b) Skewed left (c) Skewed right

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Weighted Mean

Suppose your midterm test score is 83 and your final exam score is 95.

Using weights of 40% for the midterm and 60% for the final exam, compute the weighted average of your scores.

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Solution

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Harmonic Mean

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Geometric Mean

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Example 1 – In the calculatorBelleview College must make a report to the budget committee about the average credit hour load a full-time students carries. (A 12-credit-hour load is minimum requirement for full-time status. For the same tuition, students may take up to 20 credit hours.) A random sample of 40 students yielded the following information (in credit hours):

17 12 17 12 18 19 12

12 14 15 15 12 13 14

15 14 16 16 20 13 15

18 16 17 12 12 17 15

12 15 13 18 20 12

17 16 14 19 13 12

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Example 2Barron’s Profiles of American Colleges, 19th edition, lists average class size for introductory lecture courses at each of the profiled institutions. A sample of 20 colleges and universities in California showed class sizes for introductory lecture courses to be:

14 20 20 20 20 23 25 30 30

30 35 35 35 40 40 42 50 50

80 80

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