Central Tendency

Roughly describes where the center of the data is in the set.

2 Important measures of center: a) Mean b) Median

Center

Mean

Sample Population

Mean =

n

xx

N

x

Example: Find the mean of the following sample.

23, 25, 26, 29, 39, 42, 50

4.337

2347

50423929262523

x

n

xx

Show Your Work – Including the Formula that you used!

The 50 states plus the District of Columbia have a total of 3137 counties. There are a total of 248,709,873 people in each of these counties. Find the average population per county.

Example:

countyperresidents

N

x

7.282,79

3137

873,709,248

What if I used the Almanac Book of Facts and chose a few samples? Find the means.

Sample 1 Sample 2 Sample 3

20,095 28,895 16,934

108,978 10,032 519

15,384 16,174 73,478

13,931 959,275 14,798

24,960 30,797 13,859

6.669,36x 6.034,209x 6.917,23x

Do you suppose these are close to the values we’d get if we could use the population?

We will study this further later on to see how to be able to use samples to predict the populations better.

What Exactly is the Mean?

The mean tells us how large each observation in the data set would be if the total were split equally among all the observations.

A group of elementary school children was asked how many pets they have. Here are there responses. Find the mean and explain what it means.

1 3 4 4 4 5 7 8 9

Example

petsx 5 We can look at it this way:If every child in the group had the same number of pets, each would have 5 pets.

In the last unit, we introduced the median as an informal measure of center that described the “midpoint” of a distribution.

Now, it is time to offer an official “rule” for calculating the median.

Median

The median is the value in the middle!

50% of the data is above and below this value.

Steps: 1. Put numbers in order. 2. If the # of numbers is odd – median is the middle number. 3. If the # of numbers is even – median is the average of the two #’s in the middle.

Median

Find the median.

17,14,12,8,6

Median = 12

28,27,23,22,15,7

Median=

Median = 22.5

Example. The stemplot shows travel times to work for New York workers. Find and interpret the median.

58

7

5006

5

5004

003

50002

5555001

50 .min4554: Key

min5.222

2520 M

“In the sample of New York workers, about half of the people reported traveling less than 22.5 minutes to work, and about half reported traveling more.”

88M

Example: Find the mean and median of the following test grades.

20 97 93 84 71

85 87 94 88 76

92 88 98 89 60

Stemplot:

847329

9887548

617

06

5

4

3

02

5.8115

1222

N

x

Comparing the Mean and Median

The median travel time is 20 minutes. The mean travel time is higher, 22.5

minutes. The mean is pulled toward the right

tail of this right-skewed distribution. The median, unlike the mean, is

resistant. If the longest travel time were 600

minutes rather than 60 minutes, the mean would get higher, but the median would not change at all!

Take a Look at the Stemplot and let’s discuss the mean and the median.

06

5

004

003

5002

5200001

50

Mean = 22.5Median = 20

The mean and median of a roughly symmetric distribution are close together.

If the distribution is exactly symmetric, the mean and median are exactly the same.

In a skewed distribution, the mean is usually farther out in the long tail than is the median.

In General

The mean is greatly affected by outliers – it is very sensitive to them – which means it is pulled towards the outlier.

The median is insensitive to outliers. It is often used more because it is more stable.

Salaries for MLB players or NFL players

Scores on a test when there’s one that has NOT been made up yet.

Home prices

Personal Incomes

College tuitions

Examples of where median is the best choice

Graphical Representations

Based only on the stemplot, would you expect the mean travel time to be less than, about the same as, or larger than the median? Why?

Use the stemplot to answer the following questions.

58

7

5006

5

5004

003

50002

5555001

50

Since the distribution is skewed to the right, we would expect the mean to be larger than the median.

Use your calculator to find the mean and median travel time. Was your answer to Question 1 correct?

Use the stemplot to answer the following questions.

58

7

5006

5

5004

003

50002

5555001

50

The mean is 31.25 minutes, which is bigger than the median of 22.5 minutes.

Interpret your result from Question 2 in context without using the words “mean” or “average.”

Use the stemplot to answer the following questions.

58

7

5006

5

5004

003

50002

5555001

50

If we divided the travel time up evenly among all 20 people, each would have a 31.25 minute travel time.

Would the mean or median be a more appropriate summary of the center of this distribution of drive times? Justify your answer.

Use the stemplot to answer the following questions.

58

7

5006

5

5004

003

50002

5555001

50

Since the distribution is skewed, the median would be a better measure of the center.

M F F F M F M F M M F F F M

Proportion of Success:

14

8

#

p

n

successesofp

For a Population use “P”

Trimmed Mean A disadvantage of the mean is that it can be affected by extremely high or low values.

One way to make the mean more resistant to exceptional values and still sensitive to specific data values is to do a trimmed mean.

Order the data – delete a selected number of values from each end of the list then average the remaining values.

Trimming Percentage: The percent of values trimmed from the list.

Trimmed Mean

Example80805050424040353535

30303025232020202014

a) Compute the mean for the entire sample.

b) Compute a 5% trimmed mean.

0.3620

719

n

xx

.8014

.

1

105.020:%5

ANDREMOVE

SETTHEOFBOTTOMANDTOP

THEFROMVALUEREMOVE

Trim

7.3418

625

n

xx

Example80805050424040353535

30303025232020202014

c) Compute the median for the entire sample.

d) Compute a 5% trimmed median.

5.322

3530 Median

The median is still 32.5.

e) Is the trimmed mean or the original mean closer to the median?

Trimmed Mean

Ed took 5 tests and his average was 85. If his average after the first three tests was 83, what’s the average of the last two tests?

TestsAllofSumTotalx

x

x

425

5855

85

TestsstofSumTotalx

x

x

31249

3833

83

TestslastofSumx

x

2176

249425

882

176

x

x

On Thursday, 20 out of 25 students took a test and their average was 80. On Friday, the other 5 students took it and their average was 90. What was the class average?

82

25

9058020

The first 3 hours of a trip, Susan drove 50 mph. Due to delays, she drove 40 mph for the next 2 hours. What was her average speed?

mph

hrs

mphhrsmphhrs46

.5

40.250.3

Ed’s average on 4 tests is 80. What does he need to get on the 5th test to raise his average to an 84?

320

4

480

x

scoresofx

100

4203205

32084

x

x

x