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Section 2.1 ~ Data Types and Levels of Measurement Introduction to Probability and Statistics Ms. Young

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Objective Sec. 2.1 To be able to classify data as qualitative or quantitative, to identify quantitative data as discrete or continuous, and to assign data at a level of measurement (nominal, ordinal, interval, or ratio). Why is this important? Learning how to classify the data will help us in later chapters with summarizing and displaying the data. It is very important to give a representative display of the data to avoid misconceptions.

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Data Types Two basic types: qualitative and quantitative Qualitative (categorical) data – values that can be placed in non-numerical categories Examples ~ eye color, ice cream flavors, car models, ratings (movie ratings, letter grades, pain scale ratings, etc.), social security numbers, etc. Numbers that wouldnt be used for computations would be considered qualitative Quantitative data – consist of values representing counts or measurements Examples ~ times of runners in a race, incomes of college graduates, the number of students in different classes, temperature, etc. Sec. 2.1

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Example 1 Classify each of the following sets of data as either qualitative or quantitative Brand names of shoes in a consumer survey Qualitative; brands are categorical Scores on a multiple choice exam Quantitative; the numbers represent a count of how many questions were right Letter grades on an essay assignment Qualitative; letter grades categorize based on ability level Numbers on uniforms that identify basketball players Qualitative; the numbers identify the player, but wouldnt be used to make computations Sec. 2.1

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Quantitative Data Can be classified as either continuous or discrete Continuous data – data that can take on any value in a given interval Can be part of a whole number Examples ~ Weight; someone can weigh lbs Time; the time can be part of an hour (12:48 pm) Distance; length can be measured in parts (2.36 miles) Money; can be part of a dollar ($5.84) Sec. 2.1

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Quantitative Data Contd.. Sec. 2.1 Discrete data – data that can take on only particular values and not others in-between Examples ~ Number of students (whole numbers only) Shoe sizes (whole and half sizes only) Number of times a student took their drivers test (whole numbers only)

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Example 2 For each data set, indicate whether the data are discrete or continuous Measurements of the time it takes to walk a mile Continuous The numbers of calendar years (such as 2007, 2008, 2009, etc.) Discrete The numbers of dairy cows on different farms Discrete The amounts of milk produced by dairy cows on a farm Continuous Sec. 2.1

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Qualitative Levels of Measurement A level of measurement is a further classification of qualitative or quantitative data Qualitative (categorical) data can be classified further as either nominal or ordinal Nominal level of measurement – characterized by data that consist of names, labels, or categories only; cannot be ranked or ordered Examples ~ eye color, ice cream flavors, jersey numbers, and gender of animals Ordinal level of measurement – qualitative data that can be arranged in some order (such as low to high or high to low) Examples ~ star ratings of movies, pain level ratings, letter grades on a test, etc. Time out to Think on P.56 Sec. 2.1

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Quantitative Levels of Measurement Quantitative (numerical) data, whether its discrete or continuous, can be further classified as interval or ratio Interval level of measurement – applies to quantitative data in which intervals (difference) are meaningful and ratios (which involve division) are not Data at this level have do not have a true zero point Example ~ Temperature; Intervals (differences) are meaningful - 81˚F is hotter than 80˚F by the same amount that 28˚F is hotter than 27˚F Ratios (division) are not meaningful - 20˚F is not twice as hot as 10˚F because its zero point (0˚F) does not represent a state of no heat Sec. 2.1

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Quantitative Levels of Measurement Contd… Ratio level of measurement – applies to quantitative data in which both intervals and ratios are meaningful Data at this level have a true zero point Example ~ Walking distance; Intervals (differences are meaningful) - 10 miles is further than 5 miles by the same amount that 20 miles is further than 15 Ratios (division) are meaningful - 10 miles really is twice as far as 5 miles because there is a true zero when it comes to distance Other examples that would be classified as a ratio level of measurement include, but are not limited to, weights, speeds, and incomes Sec. 2.1

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Example 3 Identify the level of measurement for each of the following sets of data Numbers on uniforms that identify players on a basketball team Nominal; the numbers are not numerically significant making these data qualitative and furthermore, the order is not meaningful. A player with the number 10 is not necessarily any better than a player with the number 3 on their jersey Student rankings of cafeteria food as excellent, good, fair, or poor Ordinal; there is no numerical significance and the categories are ranked from high to low Calendar years of historical events, such as 1776, 1945, or 2001 Interval; this is quantitative because the numbers are meaningful, but they are only meaningful when it comes to differences. They are not significant at the ratio level because there is no true zero. The year 0 is not the beginning of time Sec. 2.1

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Example 3 Contd… Identify the level of measurement for each of the following sets of data Temperatures on the Celsius scale Interval; the differences are meaningful, but ratios are not. There is no true zero point – 0 degrees Celsius does not represent a state of no heat. It would not be appropriate to say that -40 degrees Celsius is twice as cold as -20 degrees Celsius. Runners times in the Boston Marathon Ratio; time has a true zero. A time of 0 hours is the start of the race and it would be meaningful to say that 6 hours really is twice as long as 3 hours. Sec. 2.1

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Summary Sec. 2.1 Data can be classified as qualitative or quantitative Qualitative data can be classified as nominal or ordinal Quantitative data can be classified as discrete or continuous and further as interval or ratio Qualitative Quantitative Nominal Ordinal Discrete Continuous Interval Ratio Interval Ratio

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