2.3 Uses of Percentages in Statistics LEARNING GOAL Understand how percentages are used to report statistical results and recognize ways in which they are sometimes misused. Page 11

Conversions Between Fractions and Percentages To convert a percentage to a common fraction: Replace the % symbol with division by 100; simplify the fraction if necessary. Example: 25% = = To convert a percentage to a decimal: Drop the % symbol and move the decimal point two places to the left (that is, divide by 100). Example: 25% = 0.25 25 100 1 4 Page 12

Conversions Between Fractions and Percentages To convert a decimal to a percentage: Move the decimal point two places to the right (that is, multiply by 100) and add the % symbol. Example: 0.43 = 43% To convert a common fraction to a percentage: First convert the common fraction to a decimal; then convert the decimal to a percentage. Example: = 0.2 = 20% Page 12 1 5

EXAMPLE 1 Newspaper Survey 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? Solution: The 44% represents the fraction of respondents who said the President is doing a good job. Because “of” usually indicates multiplication, we multiply: 44% × 1,069 = 0.44 × 1,069 = 470.36 ≈ 470 About 470 out of the 1,069 people said the President is doing a good job. Note that we round the answer to 470 to obtain a whole number of people. (The symbol ≈ means “approximately equal to.”) Page 12 Slide 2.3- 4

absolute change = new value – reference value Using Percentages to Describe Change Absolute and Relative Change The absolute change describes the actual increase or decrease from a reference value to a new value: absolute change = new value – reference value The relative change describes the size of the absolute change in comparison to the reference value and can be expressed as a percentage: relative change = × 100% Page 12 new value – reference value reference value Slide 2.3- 5

TIME OUT TO THINK Compare the formulas for absolute and relative change to the formulas for absolute and relative error, given in Section 2.2. Briefly describe the similarities you notice. Page 69. Note that the formulas for absolute and relative error are on page 62. Slide 2.3- 6

EXAMPLE 2 World Population Growth 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. Solution: The reference value is the 1950 population of 2.6 billion and the new value is the 2000 population of 6.0 billion. absolute change = new value – reference value = 6.0 billion – 2.6 billion = 3.4 billion Page 12 Slide 2.3- 7

EXAMPLE 2 World Population Growth Solution: (cont.) relative change = × 100% = × 100% = 130.7% World population increased by 3.4 billion people, or by about 131%, from 1950 to 2000. new value – reference value reference value 6.0 billion – 2.6 billion 2.6 billion Page 12 Slide 2.3- 8

By the Way ... According to United Nations and U.S. Census Bureau estimates, world population passed 6 billion in late 1999—only 12 years after passing the 5-billion mark. World population reached 6.5 billion in early 2006 and will probably reach the 7-billion mark by about 2012. Page 69 Slide 2.3- 9

Using Percentages for Comparisons Percentages are also commonly used to compare two numbers. In this case, the two numbers are the reference value and the compared value. • The reference value is the number that we are using as the basis for a comparison. • The compared value is the other number, which we compare to the reference value. Page 12 Slide 2.3- 10

Absolute and Relative Difference The absolute difference is the difference between the compared value and the reference value: absolute difference = compared value - reference value The relative difference describes the size of the absolute difference in comparison to the reference value and can be expressed as a percentage: relative difference = ×100% page 70. (See Section 6.4 for a discussion of the meaning of life expectancy.) compared value - reference value reference value Slide 2.3- 11

absolute difference = compared value - reference value EXAMPLE 3 Russian and American Life Expectancy 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. Solution: We are told to compare the American male life expectancy to the Russian male life expectancy, which means that we use the Russian male life expectancy as the reference value and the American male life expectancy as the compared value: absolute difference = compared value - reference value = 75 years – 59 years = 16 years Page 12 Slide 2.3- 12

compared value - reference value EXAMPLE 3 Russian and American Life Expectancy Solution: (cont.) relative difference = × 100% = × 100% = 27% The life expectancy of American men is 16 years greater in absolute terms and 27% greater in relative terms than the life expectancy of Russian men. compared value - reference value reference value 75 years – 59 years 59 years Page 70 Slide 2.3- 13

“Of” versus “More Than” (or “Less Than”) • If the new or compared value is P% more than the reference value, then it is (100 + P)% 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. Page 71 Slide 2.3- 14

EXAMPLE 4 World Population 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. Solution World population in 2000 was 131% more than world population in 1950. Because (100 + 131)% = 231%, the 2000 population was 231% of the 1950 population. This means that the 2000 population was 2.31 times the 1950 population. Page 12 Slide 2.3- 15

TIME OUT TO THINK One store advertises “1/3 off everything!” Another store advertises “Sale prices just 1/3 of original prices!” Which store is having the bigger sale? Explain. Page 71 See also Example 5. Slide 2.3- 16

Percentages of Percentages Percentage Points versus % When you see a change or difference expressed in percentage points, you can assume it is an absolute change or difference. If it is expressed as a percentage, it probably is a relative change or difference. Page 71 See also Example 5. Slide 2.3- 17

EXAMPLE 6 Margin of Error 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. Solution The range of 20% to 30% comes from subtracting and adding an absolute difference of 5 percentage points to 25%. That is, 20% = (25 – 5)% and 30% = (25 + 5)% Therefore, the correct statement is “25% with a margin of error of 5 percentage points.” If you instead said “25% with a margin of error of 5%,” you would imply that the error was 5% of 25%, which is only 1.25%. Page 72 (See Section 1.1 to review the meaning of “margin of error.”) Slide 2.3- 18

The End Page 72 (See Section 1.1 to review the meaning of margin of error.) Slide 2.3- 19