2.3: Measures of Central Tendency

Chapter 2: Descriptive Statistics

Objectives . . . • Determine the mean, median, and mode of a population and of a sample• Determine the weighted mean of a data set and the mean of a frequency

distribution• Describe the shape of a distribution as symmetric, uniform, or skewed and

compare the mean and median for each

Measure of central tendency• A value that represents a typical, or central, entry of a data set.• Most common measures of central tendency:

1)__________________ 2) __________________ 3) _________________

1) Mean (average) – _______ of data entries _____________ by the

____________ of _____________.• Population mean:

• Sample mean:

Example: Finding a Sample Mean

The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. What is the mean price of the flights?

872 432 397 427 388 782 397

Round-off Rules: mean has one more decimal place than set of data values. Rounding should not be done until final answer of calculation.

2) Median - The value that lies in the ____________ of the data when the

data set is _____________.

• Measures the _____________________________________________

• If the data set has an odd number of entries:

even number of entries:

Example 2A: Finding the Median

The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the median of the flight prices.

872 432 397 427 388 782 397

Example 2B: Finding the Median

The flight priced at $432 is no longer available. What is the median price of the remaining flights?

872 397 427 388 782 397

3) Mode• The data entry that occurs with the ______________

______________.• If no entry is repeated:• If two entries occur with the same greatest frequency,

Example 3A: Finding the ModeThe prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the mode of the flight prices.

872 432 397 427 388 782 397

Example 3B: Finding the Mode

At a political debate a sample of audience members was asked to name the political party to which they belong. Their responses are shown in the table. What is the mode of the responses?

Political Party Frequency, f

Democrat 34

Republican 56

Other 21

Did not respond 9

Mode is the only measure of central tendency that can be used to describe data at the

_____________ level of measurement.

When working with __________________ data the mode is ____________ used.

Comparing the Mean, Median, and Mode

• All three measures describe a typical entry of a data set.• Advantage of using the mean:

• Disadvantage of using the mean:

Example 4: Comparing the Mean, Median, and ModeFind the mean, median, and mode of the sample ages of a class shown. Which measure of central tendency best describes a typical entry of this data set? Are there any outliers?

Ages in a class

20 20 20 20 20 20 21

21 21 21 22 22 22 23

23 23 23 24 24 65

Weighted Mean• The mean of a data set whose entries have varying weights.•

Example 5: Finding a Weighted MeanIn this class class your grade is determined by two separate weighted averages: 90% for your trimester marking period grade, and 10% for your final exam.Your marking period grade is determined by: 45% for Tests; 35% for Quizzes, and 20% for Homework.

Lets say that you have a 75% in homework, an 86% on quizzes and a 82% on tests. What is your trimester marking period grade?

If you get a 100% on the final can you get a B+?

Mean of a Frequency Distribution• Approximated by

where x and f are the midpoints and frequencies of a class, respectively, and n is

Finding the Mean of a Frequency Distribution

In Words In Symbols

1. Find the midpoint of each class

2. Find the sum of the products of the midpoints and the frequencies

3. Find the sum of the frequencies (sample size)

4. Find the mean of the distribution

Example: Find the Mean of a Frequency Distribution

Use the frequency distribution to approximate the mean number of min. that a sample of Internet subscribers spent online in their most recent session.

Class Midpoint Freq. f

7 – 18 6

19 – 30 10

31 – 42 13

43 – 54 8

55 – 66 5

67 – 78 6

79 – 90 2

n =

_______________ Distribution• A vertical line can be drawn through the middle of a graph of the

distribution and the resulting halves are approximately mirror images.

The Shape of Distributions

_____________ Distribution (rectangular)• All entries or classes in the distribution have equal or

approximately equal frequencies.• Symmetric.

The Shape of Distributions Contd.

__________ ____________ Distribution (negatively skewed)

• The “tail” of the graph elongates more to the __________.

• The mean is to the ____________ of the median.

___________ ____________ Distribution (positively skewed)

• The “tail” of the graph elongates more to the ___________.

• The mean is to the ___________ of the median.