Section 2.3 ~ Uses of Percentages in Statistics

Introduction to Probability and StatisticsMs. Young

Objective

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To understand how percentages are used to report statistical results and recognize ways in which they are sometimes misused.

Why is this important?To understand the true meaning of statements that involve percentages which helps to make educated decisions and fully understand statistical statements.

“The rate of smoking among 10th graders jumped 45%, to 18.3%, and the rate for 8th graders is up 44%, to 10.4%

The Basics of Percentages A percentage is simply a way to represent

a fraction; part per 100 Conversions Between Fractions and

Percentages: Percent to Fraction:

Take the percent and write it out of 100, then reduce the fraction

Ex. ~

Percent to Decimal: Drop the % symbol and move the decimal point two

places to the left (that is, divide by 100) Ex. ~

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25 125%

100 4

25% .25

The Basics of Percentages Cont’d…

Decimal to percent: Move the decimal point two places to the right (that

is, multiply by 100) and add the % sign Ex. ~

Fraction to percent: Convert the fraction to a decimal, then convert the

decimal to a percent Ex. ~

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.43 43%

1.2 20%

5

Example 1 A newspaper reports that 44% of 1,069 people

surveyed said that the President is doing a good job. How many people said that the President is doing a good job? “Of” is a common word used for multiplication 44% of 1,069 would be equivalent to:

About 470 out of the 1,069 people said the President is doing a good job

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069,144. 47036.470

Using Percentages to Describe Change Percentages are commonly used in statistics to

describe how data change with time (absolute change and relative (percent) change) Ex. ~ The population of a town was 10,000 in 1970 and

15,000 in 2000 When calculating change, you are always

dealing with two values: the starting point, or reference value, and a new value that is either an increase or a decrease in comparison to the reference value Ex. ~ using the case above, the reference value

would be the 10,000 people and the new value would be the 15,000 people

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Absolute Change Absolute change – describes the actual

increase or decrease from a reference value to a new value:

Example ~ The population of a town was 10,000 in 1970 and 15,000 in 2000. The absolute change is:

A positive absolute change represents an increase from the original value

A negative absolute change represents a decrease from the original value

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absolute change = new value - reference value

Absolute change = 15,000 - 10,000 = 5,000 people

Relative Change

Relative change – describes the size of absolute change in comparison to the reference value (original value) and is expressed as a percentage

Example: The population of a town was 10,000 in 1970 and 15,000 in 2000. The relative change is:

A positive relative change represents a percent increase from the original value

A negative relative change represents a percent decrease from the original value

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absolute change

new value - reference valueRelative change = 100%

reference value

15,000 - 10,000Relative change = 100%

10,000

5,000Relative change = 100% 50% increase

10,000

Example 2 World population in 1950 was 2.6 billion. By the beginning

of 2000, it had reached 6.0 billion. Describe the absolute and relative change in world population from 1950 to 2000. The reference value is 2.6 billion The new value is 6.0 billion The absolute change is:

The population increased by 3.4 billion people from 1950 to 2000

The relative change is:

The population increased by about 131% from 1950 to 2000

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absolute change = 6.0 billion - 2.6 billion = 3.4 billion

6.0 billion - 2.6 billionRelative change = 100%

2.6 billion

3.4 billionRelative change = 100% 130.8%

2.6 billion

Using Percentages for Comparisons

Similar formulas are used to make comparisons between two numbers that are not necessarily a change in time. This is known as absolute and relative difference. Ex. ~ the number of hours a woman is in labor with her

second child in comparison to her first child The numbers that are being compared are

classified as: The reference value – the number that is being used

as the basis for a comparison In the case above, the number of hours the woman was in

labor with her first child would be the reference value The compared value – the other number that is being

compared to the reference value The number of hours the woman was in labor with her

second child would be the compared value

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Absolute Difference Absolute difference – the difference between the

compared value and the reference value

Ex. ~ Sue was in labor with her first child for 22 hours and was in labor with her second child for 8 hours. The absolute difference would be:

This means that she was in labor for 14 hours less with her second child

A positive absolute difference represents an increase in comparison to the reference value

A negative absolute difference represents a decrease in comparison to the reference value

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absolute difference = compared value - reference value

absolute difference = 8 - 22 = -14 hours

Relative Difference Relative difference – describes the size of the

absolute difference in comparison to the reference value and is expressed as a percentage

Ex. ~ Sue was in labor with her first child for 22 hours and was in labor with her second child for 8 hours. The relative difference would be:

This means that she was in labor for 63% less time with her second child

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Absolute difference

compared value - reference valueRelative difference = 100%

reference value

8 - 22Relative difference = 100 .63 63%

22

Example 3 Life expectancy for American men is about 75 years, while

life expectancy for Russian men is about 59 years. Compare the life expectancy of American men to that of Russian men in absolute and relative terms. The reference value is the life expectancy of Russian men The compared value is the life expectancy of American men The absolute difference is:

This means that American men can expect to live about 16 years longer than Russian men

The relative difference is:

This means that American men can expect to live about 27% longer than Russian men

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absolute difference = 75 years - 59 years = 16 years

75 years - 59 yearsRelative difference = 100% .2712 27%

59 years

Of versus More Than (or Less Than)

There are two equivalent ways to state change in terms of percentages: Ex ~ Suppose an item is on sale for 10% off its original

price. One way to explain this is by using the phrase less than:

“The sale price is 10% less than the original price” Another way to explain this is by using the phrase of:

“The sale price is 90% of the original price” Since the original price is 100%, the sale price would be

90% of the original price (100% - 10%)

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Of versus More Than (or less than) in general

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If the new or compared value is P% more than the reference value, then it is (100 + P)% of the reference value Ex. ~ 40% more than the reference value would be

140% of the reference value

If the new or compared value is P% less than the reference value, then it is (100 - P)% of the reference value Ex. ~ 40% less than the reference value would be 60%

of the reference value

Example 4 In Example 2, we found that world population in

2000 was about 131% more than world population in 1950. Express this change with an “of ” statement. Since the population in 1950 is the reference value and

the population is 2000 is 131% more than that reference value, the population can be expressed by saying:

The population in 2000 is 231% (100% + 131%) of the population in 1950

In other words, the population in 2000 is 2.31 times the population in 1950

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Example 5 A store is having a “25% off” sale. In general,

how does a sale price compare to an original price? If the original price is $30 what is the sale price? In general, the sale price is 25% less than the original

price or 75% of the original price (100% - 25%) If the original price is $30, then the sale price is:

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75% $30 .75 30 $22.50

Percentages of Percentages Percent changes and percent differences can be particularly

confusing when the values themselves are percentages Ex. ~ Your bank increases the interest rate on your savings

account from 3% to 4%. You most likely want to say that it was a 1% increase, when in reality it was a 33% increase

The 1% is the absolute change expressed as a change in percentage points

So it would be accurate to say that your savings account increased 1 percentage point

The 33% is the relative change expressed as a percent change This value is found by taking the new value and comparing it to the old value

When you see a change or difference expressed in percentage points, you can assume it is an absolute change or difference

When you see a change or difference expressed as a percent, you can assume it is a relative change or difference

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4% - 3%Relative change = 100% 33% increase

3%

Example 6 Based on interviews with a sample of students at

your school, you conclude that the percentage of all students who are vegetarians is probably between 20% and 30%. Should you report your result as “25% with a margin of error of 5%” or as “25% with a margin of error of 5 percentage points”? Explain. It should be reported as 25% with a margin of error of 5

percentage points If you said 25% with a margin of error of 5% that would

be a relative change and would really refer to an interval between 23.75% and 26.25%

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