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KU Math Center Sunday, Wednesday & Thursday: 8:00pm-12:00am (midnight) Monday: 11:00am-5:00pm AND Tuesday: 11:00am-12:00am (midnight) All times are Eastern Time at: http://khe2.acrobat.com/mathtutor/ Additional Information about the Math Center is in the Doc Sharing Portion of the course. Can't Type? press F11 or F5. Can’t Hear? Check: Speakers, Volume or Re-Enter. Put "?" in front of Questions so it is easier to see them. Tech support: 866.348.1196

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Data Types and Levels of Measurement 2.1 Data Types and Levels of Measurement

Data Types Quantitative data Quantitative data has a value or a numerical measurement for which you can calculate sums, products and other numerical calculations. You can do meaningful math Qualitative data Qualitative data is grouped into a category or group. Sums, products or other numerical calculations do not mean anything. You cannot do meaningful math

Brand names of shoes in a consumer survey EXAMPLE 1 Data Types Classify each of the following sets of data as qualitative or quantitative. Brand names of shoes in a consumer survey Scores on a multiple-choice exam Solution: Brand names are categorical and therefore represent qualitative data. Scores on a multiple-choice exam are quantitative because they represent a count of the number of correct answers.

Quantitative Data Continuous data can take on any value in a given interval Age: between 10 years and 11 years old there can be any number between, 10.4, 10.45, 10.455, 10.4555, 10.89, 10.9999 Discrete data can take on only particular, distinct values and not other values in between Number of children in a family must be and will only be a whole number

EXAMPLE 2 Discrete or Continuous? For each data set, indicate whether they data are discrete or continuous. Measurements of the time it takes to walk a mile The number of calendar years (such as 2007, 2008, 2009) Solution: Time can take on any value, so measurements of time are continuous. The number of calendar years are discrete because they cannot have fractional values. For example, on New Years Eve of 2009, the year will change from 2009 to 2010; we’ll never say the year is 2009½ .

Levels of Measurement Nominal: Data is put in categories [names] Ordinal: Nominal, plus the data is put in ordered categories [ranks] Interval: Ordinal, plus the intervals are equal, but ratios are not meaningful [arbitrary zero] Ratio: Interval, plus the data have an absolute zero point [ratios are meaningful]

Identify the level of measurement (nominal, ordinal, interval, ratio) for each of the following sets of data. Numbers on uniform that identify players on a basketball team Student rankings of cafeteria food as excellent, good, fair, or poor Calendar years of historic events, such as 1776, 1945, or 2001 Temperatures on the Celsius scale Runners’ times in the Boston Marathon

2.2 Dealing with Errors

Two Types of Measurement Error Random errors occur because of random and inherently unpredictable events in the measurement process. Systematic errors occur when there is a problem in the measurement system that affects all measurements in the same way.

Absolute and Relative Errors The absolute error describes how far a claimed or measured value lies from the true value: absolute error = claimed or measured value – true value The relative error compares the size of the absolute error to the true value. It is often expressed as a percentage: relative error = x 100% absolute error true value

Find the absolute and relative error. Your true weight is 100 pounds, but a scale says you weigh 102 pounds. Solution: The measured value is the scale reading of 105 pounds and the true value is 100 pounds. absolute error = measured value – true value = 102 lbs – 100 lbs = 2 lbs relative error = absolute error / true value x 100% 2 lbs/100 lbs. x 100% = 2%

Describing Results Accuracy describes how closely a measurement approximates a true value. An accurate measurement is close to the true value. (Close is generally defined as a small relative error, rather than a small absolute error.) Precision describes the amount of detail in a measurement. You can precisely measure something inaccurately.

Suppose that your true weight is 102. 4 pounds Suppose that your true weight is 102.4 pounds. The scale at the doctor’s office, which can be read only to the nearest quarter pound, says that you weigh 102¼ pounds. The scale at the gym, which gives a digital readout to the nearest 0.1 pound, says that you weigh 100.7 pounds. Which scale is more precise? Which is more accurate? Solution: The scale at the gym is more precise because it gives your weight to the nearest tenth of a pound, whereas the doctor’s scale gives your weight only to the nearest quarter pound. However, the scale at the doctor’s office is more accurate because its value is closer to your true weight.

Uses of Percentages in Statistics 2.3 Uses of Percentages in Statistics

EXAMPLE 1 Newspaper Survey A newspaper reports that 43% of 1,070 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 43% represents the fraction of respondents who said the President is doing a good job. Because “of” usually indicates multiplication, we multiply: 43% × 1,070 = 0.43 × 1,070 = 460.1 ≈ 460 About 460 out of the 1,070 people said the President is doing a good job. Note that we round the answer to 460 to obtain a whole number of people. (The symbol ≈ means “approximately equal to.”)

absolute change = new value – reference value 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% new value – reference value reference value

World population in 1950 was 2. 5 billion World population in 1950 was 2.5 billion. By the beginning of 2000, it had reached 7.0 billion. Describe the absolute and relative change in world population from 1950 to 2000. Solution: The reference value is the 1950 population of 2.5 billion and the new value is the 2000 population of 7.0 billion. absolute change = new value – reference value = 7.0 billion – 2.5 billion = 4.5 billion relative change = 7 billion – 2.5 billion * 100% 2.5 billion = 180.0% World population increased by 4.5 billion people, or by about 180%, from 1950 to 2000.

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.

Absolute and Relative Differences 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 = absolute difference x 100% reference value

“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. 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.

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% points.

2.4 Index Numbers

Index Numbers An index number provides a simple way to compare measurements made at different times or in different places. The value at one particular time (or place) must be chosen as the reference value (or base value). The index number for any other time (or place) is index number = x 100 value reference value

EXAMPLE 1 Finding an Index Number Suppose the cost of gasoline today is $3.79 per gallon. Using the 1975 price as the reference value, find the price index number for gasoline today. Solution: Table 2.1 shows that the price of gas was 56.7¢, or $0.567, per gallon in 1975. If we use the 1975 price as the reference value and the price today is $3.79, the index number for gasoline today is index number = × 100 = × 100 = 668.4 This index number for the current price is 668.4, which means the current gasoline price is 668.4% of the 1975 price. current price 1975 price $3.79 $0.567

The Consumer Price Index The Consumer Price Index (CPI), which is computed and reported monthly, is based on prices in a sample of more than 60,000 goods, services, and housing costs.

We compare the Consumer Price Indices for 2006 and 2000: EXAMPLE 3 CPI Change Suppose you needed $8,350 to maintain a particular standard of living in 2000. How much would you have needed in 2006 to maintain the same living standard? Solution: We compare the Consumer Price Indices for 2006 and 2000: CPI for 2006 CPI for 2000 = = 1.17 201.6 172.2 That is, typical prices in 2006 were about 1.17 times those in 2000. So if you needed $ 8,350 in 2000, you would have needed 1.17 x $ 8,350 = $9,770 to have the same standard of living in 2006. (The true value is 9,800, from http://aspe.hhs.gov/poverty/06poverty.shtml)