Section 2.3 ~ Uses of Percentages in Statistics

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Section 2.3 Uses of Percentages in Statistics

• Introduction to Probability and Statistics
• Ms. Young

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Objective
Sec. 2.3
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
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The Basics of Percentages
Sec. 2.3

• 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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The Basics of Percentages Contd
Sec. 2.3

• 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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Example 1
Sec. 2.3

• 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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Using Percentages to Describe Change
Sec. 2.3

• 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
Sec. 2.3

• 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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Relative Change
Sec. 2.3

• 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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Example 2
Sec. 2.3

• 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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Using Percentages for Comparisons
Sec. 2.3

• 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
Sec. 2.3

• 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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Relative Difference
Sec. 2.3

• 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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Example 3
Sec. 2.3

• 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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Of versus More Than (or Less Than)
Sec. 2.3

• 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
Sec. 2.3

• 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

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Example 4
Sec. 2.3

• 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
Sec. 2.3

• 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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Percentages of Percentages
Sec. 2.3

• 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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Example 6
Sec. 2.3

• 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