2.4: Measures of CenterObjective: To calculate and interpret measures of center given sets of data

CHS Statistics

3 Types of Center Measurement

1)Mean

2)Median

3)Mode

3 Types of Center Measurement

1)MeanoDivide the sum of each datum and divide by the sample size (the number of data entries)

oThe mean is generally the most important/most utilized descriptive measurement.

3 Types of Center Measurement

1)Mean (continued)

Ex) The prices (in dollars) for a sample of round trip flights from Chicago to Cancun are listed. What is the mean price of the flights?

872 432 397 427 388

873 782 397

3 Types of Center Measurement

2) Median oThe middle value of the data set arranged in ascending (or descending) order.

oIf the number of values is odd, the median is the exact middle value.

oIf the number of values is even, the median is the mean of the two middle values.

oOften denoted as

3 Types of Center Measurement

2) Median (continued)Example: The flight priced at $432 is no

longer available. What is the median price of the remaining flights?

872 397 427 388782 397

3 Types of Center Measurement

3) Mode oThe value(s) in a set of data that occurs the most.

oIf no value is repeated, we say that is has no mode. However, one could argue that all values are modes….

oExample: Find the mode of the flight prices.

872 432 397 427 388782 397

OutliersoA value that is much higher or lower than the mean. We will discuss the rule of thumb for identifying outliers at a later date.

oAffect mean?

oAffect median?

Robert Wadlow 8 Ft 11.1 inchesBao Xishun 7 ft 9 inches

He Pingping2 ft 4 inches

OutliersExample: The President of a company makes $100,000. His 6 computer technicians make $30,000, $32,000, $35,000, $38,000, $38,000, and $42,000. The secretary makes $20,000.

• What are the mean, median, and mode of these data?

 

• What are the mean, median, and mode of these data, if the president’s salary is taken out?

Comparison of Center Measures

Name Definition

How Commo

n?Existenc

e

Takes Every

Value into Account

Affected by

Extreme Values

Advantage

Disadvantage

Mean               

 Median              

 Mode              

General Notes About Center Measures

•When data are fairly symmetric, the mean and median tend to be about the same, but the mean is usually a better measure of center.

• If the data are skewed, the median is the better measure of center.

What’s Your Height in Inches?• Input these data into L1 in your graphing calculator.–STAT Edit

•Create a histogram of these data:–2nd Statplot ON Select Histogram Select L1 for data in this case Frequency: 1–Zoom 9: ZoomStat

•Are these data skewed?

What’s Your Height in Inches?

•Depending on the shape of the distribution, find an appropriate measure of center, mean or median:

•With data still in your L1:–STAT Calc 1: One-Variable Stats

Day 1 Assignment:

•pp. 65 – 66 # 2 – 8 EVEN

Day 2: Weighted Mean•A mean which is computed with different scores assigned for different weights.

weighted

Weighted MeanYou are taking a class in which your grade is determined from five categories: 50% from your test mean, 15% from your midterm, 20% from your final exam, 10% from your computer lab work, and 5% from your homework. Your scores are 86 (test mean), 96 (midterm), 82 (final exam), 98 (computer lab), and 100 (homework). What is the weighted mean of your scores? Did you earn an A?

Mean of a Frequency Distribution

o Use the midpoints and counts to estimate the mean of the data. Why wouldn’t the mean be exact?

Steps for Calculation:

1. Find the midpoints and frequencies

2. Multiply the midpoints and frequencies together.

3. Find the total of #2 (midpoint X frequency)

4. Find the total frequency

5. Divide #3 by #4 (see formula)

Mean of a Frequency Distribution

oExample:

Class midpoints

Frequency

12.5 6

24.5 10

36.5 13

48.5 8

60.5 5

72.5 6

84.5 2

Mean of a Frequency Distribution

oUse your calculator!

oSTAT Edit oPut midpoints into L1oPut frequencies into L2

oSTAT Calc 1-Variable Stats and enter L1 (2nd 1) , L2

(2nd 2)

Day 2 Assignment:

•pp. 65 – 66 # 10, 20, 21