Section 4.1 ~ What is Average?

Introduction to Probability and StatisticsMs. Young

Objective

Sec. 4.1

In this section you will understand the difference between the three most common measures of central tendency; mean, median, and mode. You will learn how each is affected by outliers and also when it is appropriate to use a weighted mean.

Mean The mean is most commonly referred to as the average The mean is found by finding the sum of all the values in a

data set and dividing that sum by the number of values in the set

Example ~ the following values represent the weights of five wrestlers:

188 162 190 150 176 Find the average weight of these five wrestlers

The average weight is 173.2 lbs

Sec. 4.1

of all valuesmean =

total number of values

sum

188 162 190 150 176 866mean = 173.2

5 5

Median The median is the middle number in a data set To find the median, you must first arrange the

values in ascending (or descending) order A data set that has an odd number of values will have

exactly 1 value in the middle Ex ~ Find the median of the data set: 4 3 6 3 6

3 3 4 6 6 A data set that has an even number of values will have

2 values in the middle which means that the median will be in the middle of those two values (or the average of those two values)

Ex ~ Find the median of the data set: 4 3 6 3 6 5 3 3 4 5 6 6

Sec. 4.1

4 54.5

2

Mode The mode is the most common value (or group of

values) in a data set A data set may have one mode, more than one mode, or no

mode Ex ~ Find the mode in the data set: 4 3 6 10 6 3 6

Since 6 occurs the most (3 times), 6 is the mode

Ex ~ Find the mode in the data set: 4 3 6 3 10 6Since 3 and 6 occur the most (2 times each), 3

and 6 would be the mode Ex ~ Find the mode in the data set 4 3 6 2 10

Since all values only occur once, there is no mode

The mode is most commonly used for qualitative data at the nominal level since neither the mean nor the median can be found for qualitative data

Ex. ~ the brand of shoes each student is wearing in this class Can’t calculate the mean because the data isn’t numerical Can’t calculate the median because the data isn’t ordinal Can calculate the mode ~ the brand that most students are

wearing

Sec. 4.1

Rounding Rule State your answers with one more

decimal place of precision than is found in the raw data Ex ~ the mean of 2, 3, and 5 is 3.3333… We would present the

answer to the nearest tenth (3.3) since the raw data

are whole numbers

Sec. 4.1

Example 1 Eight grocery stores sell the PR energy bar for the following

prices:$1.09 $1.29 $1.29 $1.35 $1.39 $1.49 $1.59 $1.79

Find the mean, median, and mode for these prices.

Mean:

Median:$1.09 $1.29 $1.29 $1.35 $1.39 $1.49 $1.59 $1.79

Mode:$1.09 $1.29 $1.29 $1.35 $1.39 $1.49 $1.59 $1.79

Sec. 4.1

$1.09 + $1.29 + $1.29 + $1.35 + $1.39 + $1.49 + $1.59 + $1.79mean =

8

$11.28 = $1.41

8

$1.35 + $1.39 $1.37

2

Effects of Outliers An outlier in a data set is a value that is much higher or

much lower than almost all other values An outlier doesn’t affect the median or mode of a data

set since an outlier doesn’t fall in the middle of a data set and it doesn’t occur most frequently

But an outlier can greatly affect the mean of a data set since it is so extreme compared to the other values

Ex. ~ The following values represent the contract offer received for five graduating college seniors in the NBA (zero means that they didn’t receive an offer)

$0 $0 $0 $0 $3,500,000

The mean would be $700,000, but it wouldn’t be fair to say that the average senior on this basketball team received a $700,000 contract

Sec. 4.1

Comparison of Mean, Median, and Mode

Measure Definition Takes every value

into account?

Affected by Outliers?

Advantages

MEAN Yes Yes Commonly understood as the average; works well with many statistical methods

MEDIAN middle value No (except for the ordering of every number)

No (since the an outlier won’t fall in the middle of a data set)

When there are outliers finding the median may be more representative of an “average” than the mean

MODE most frequent value

No No (since an outlier won’t occur most frequently)

Most appropriate for qualitative data at the nominal level

Sec. 4.1

sum of all values

total number of values

Example 2 A track coach wants to determine an appropriate heart

rate for her athletes during their workouts. She chooses five of her best runners and asks them to wear heart monitors during a workout. In the middle of the workout, she reads the following heart rates for the five athletes: 130, 135, 140, 145, and 325. Which is a better measure of the average in this case – the mean or the median? Why? The median is a better measure because the heart rate of 325 is an

outlier and will greatly affect the average if the mean is calculated This outlier was most likely caused by a faulty monitor as a heart rate

of 325 would result in cardiac arrest!

Sec. 4.1

Weighted Means A weighted mean accounts for variation in the

relative importance of data values Each data value is assigned a weight, and the weighted mean

is calculated by the formula:

Ex. ~ Suppose your course grade is based on four tests and one final exam. Each test counts as 15% of your final grade and the final exam counts as 40% of your final grade. Your test scores are 75, 80, 84, and 88 and your final exam score is a 96. What is your final grade?

Sec. 4.1

sum of (each data value its weight)weighted mean =

sum of all weights

(75 15) + (80 15) + (84 15) + (88 15) + (96 40)weighted mean =

15 + 15 + 15 + 15 + 40

8745 = 87.45%

100

Example 3 Each quarter grade is calculated with a 80% weight on tests and quizzes

and a 20% weight on class work, homework, and class participation. Furthermore, your course grade is calculated with a 40% weight on quarter 1, a 40% weight on quarter 2, and a 20% weight on the final exam.

Suppose that Joe’s overall quiz/test average in quarter 3 was an 86% and

his overall homework/class work/ class participation grade was 95%. In quarter 4 his quiz/test average was a 92% and his homework/class work/class participation grade was a 90%, and he scored an 80% on his final exam. What was Joe’s final grade?

Start by finding each quarter grade:

Now calculate the overall grade:

Sec. 4.1

(86 80) + (95 20) 8780Q1 = 87.8%

100 100

(92 80) + (90 20) 9200Q2 = 92%

100 100

(87.8 40) + (92 40) + (80 20) 8792Final grade = 87.92%

100 100

Example 4 Each quarter grade is calculated with a 80% weight on tests and quizzes

and a 20% weight on class work, homework, and class participation. Furthermore, your course grade is calculated with a 40% weight on quarter 1, a 40% weight on quarter 2, and a 20% weight on the final exam.

If Amy got an 88% for quarter 1 and a 93% for quarter 2, what would she need to get on the final exam in order to receive at least a 90% for her final grade?

Sec. 4.1

(88 40) (93 40) 2090

100

x

7240 2090

100

x

7240 2100 100

0 90

100

x

7240 20 9000x 20 1760x 88%x

Mathematical Notation Describing Means

Algebraic Form Definition

Sum of all values

Mean of a sample

Mean of an entire population

Weighted Mean

Sec. 4.1

x

x

( )

;

x w

w

x each value w weight