Chapter 9 Collecting and Interpreting Data

Populations and Samples Population: The set of objects being studiedPopulation: The set of objects being studied A population can consist of:A population can consist of: People or animals People or animals Plants Plants Inanimate objects Inanimate objects Events Events Elements: members of a populationElements: members of a population

Populations and Samples Variable: Any characteristic of elements of the populationVariable: Any characteristic of elements of the population Quantitative: a variable that is numerical Quantitative: a variable that is numerical Qualitative: a variable that is not numerical Qualitative: a variable that is not numerical

Populations and Samples, cont’d Census: measures a variable for every element of the population.Census: measures a variable for every element of the population. Is time-consuming and expensive Is time-consuming and expensive Instead of the entire population, a subset, called a sample, is usually studied.Instead of the entire population, a subset, called a sample, is usually studied.

Example 1 You want to determine voter opinion on a measure. You survey potential voters among pedestrians on Main Street during lunch.You want to determine voter opinion on a measure. You survey potential voters among pedestrians on Main Street during lunch. a) What is the population? b) What is the sample? c) What is the variable being measured?

Example 1, cont’d a) The population: all people who plan to vote on the measure.

Example 1, cont’d b) The sample: all the people you spoke to on Main Street who intend to vote.

Example 1, cont’d c) The variable: the voter’s intent to vote “yes” or “no” on the measure.

Data Data: information recorded from a sampleData: information recorded from a sample Quantitative data: measurements for a quantitative variable. Quantitative data: measurements for a quantitative variable. Qualitative data: measurements for a qualitative variable. Qualitative data: measurements for a qualitative variable.

Data, cont’d Ordinal Data: Qualitative data with a natural orderOrdinal Data: Qualitative data with a natural order For example, ranking a pizza on a scale of “Excellent” to “Poor” is ordinal. For example, ranking a pizza on a scale of “Excellent” to “Poor” is ordinal. Nominal Data: Qualitative data without a natural orderNominal Data: Qualitative data without a natural order For example, eye color is nominal. For example, eye color is nominal.

Data, cont’d The types of data are illustrated below.The types of data are illustrated below.

Example 2 You survey potential voters among the people on Main Street about their political affiliation, age, and opinion on the ballot measure.You survey potential voters among the people on Main Street about their political affiliation, age, and opinion on the ballot measure. Classify each variable as quantitative or qualitative.Classify each variable as quantitative or qualitative.

Example 2, cont’d Political affiliation: qualitative, nominal Political affiliation: qualitative, nominal Age: quantitative Age: quantitative Opinion on the ballot measure: qualitative, nominal Opinion on the ballot measure: qualitative, nominal

Samples Statistical inference: to make an estimation for the entire population, based on data from a sample.Statistical inference: to make an estimation for the entire population, based on data from a sample. Representative Sample: a sample that has characteristics typical of the population as a wholeRepresentative Sample: a sample that has characteristics typical of the population as a whole Bias: a flaw in the sampling that makes it more likely the sample will not be representative. Bias: a flaw in the sampling that makes it more likely the sample will not be representative.

5 Common Sources of Bias 1. Faulty sampling: The sample is not representative. 2. Faulty questions: Questions are worded to influence the answers. 3. Faulty interviewing: Interviewers fail to survey the entire sample, misread questions, or misinterpret answers.

Common Sources of Bias, cont’d 4. Lack of understanding: The person being interviewed does not understand the question or needs more information. 5. False answers: The person intentionally gives incorrect information.

Example 3 Suppose you wish to determine voter opinion on eliminating the capital gains tax. You survey potential voters on a street corner near Wall Street in New York City.Suppose you wish to determine voter opinion on eliminating the capital gains tax. You survey potential voters on a street corner near Wall Street in New York City. Identify a source of bias in this poll.Identify a source of bias in this poll.

Example 3, cont’d People who work on Wall Street would benefit from the elimination of the tax and are more likely to favor the elimination than the average voter may be.People who work on Wall Street would benefit from the elimination of the tax and are more likely to favor the elimination than the average voter may be. This is faulty sampling. This is faulty sampling.

Example 4 Suppose a car manufacturer wants to test the reliability of 1000 alternators. They will test the first 30 from the lot for defects.Suppose a car manufacturer wants to test the reliability of 1000 alternators. They will test the first 30 from the lot for defects. Identify any potential sources of bias.Identify any potential sources of bias.

Example 4, cont’d It may be that defects are either much more likely at the beginning of a production run or much less likely at the beginning. In either case, the sample would not be representative.It may be that defects are either much more likely at the beginning of a production run or much less likely at the beginning. In either case, the sample would not be representative. This is potentially faulty sampling. This is potentially faulty sampling.

Simple Random Samples Representative samples are usually chosen randomly.Representative samples are usually chosen randomly. Simple Random Sample: all samples of the same size are equally likely to be chosen.Simple Random Sample: all samples of the same size are equally likely to be chosen.

Stratified Sampling Stratified Sampling: the population is divided into 2 or more nonoverlapping subsets, each called a stratum.Stratified Sampling: the population is divided into 2 or more nonoverlapping subsets, each called a stratum. Can be less costly because the strata allow a smaller sample to be used. Can be less costly because the strata allow a smaller sample to be used.

Measures of Central Tendency Measures of Central Tendency: tell us where the center of the data set lies.Measures of Central Tendency: tell us where the center of the data set lies. The most important measures of central tendency are mean, median, and mode. The most important measures of central tendency are mean, median, and mode.

The Mean Mean: the most common type of average.Mean: the most common type of average. This is an arithmetic mean. This is an arithmetic mean. If there are N numbers in a data set, the mean is:If there are N numbers in a data set, the mean is: In other words, add the numbers together, and divide by how many there are.In other words, add the numbers together, and divide by how many there are.

The Mean, cont’d The mean of a sample is shown by which means “x-bar”.The mean of a sample is shown by which means “x-bar”. The mean of a population is denoted by μ, the Greek letter pronounced “mew”.The mean of a population is denoted by μ, the Greek letter pronounced “mew”.

Example 1 Find the mean of each data set.Find the mean of each data set. a) 1, 1, 2, 2, 3 b)1, 1, 2, 2, 11

Example 1 1.1, 1, 2, 2, =99/5= , 1, 2, 2, =1717/5=3.4

Example 2 A college graduate reads that a company with 5 employees has a mean salary of $48,000.A college graduate reads that a company with 5 employees has a mean salary of $48,000. How might this be misleading?How might this be misleading?

Example 2, cont’d One possibility is that every employee earns $48,000.One possibility is that every employee earns $48,000. Another possibility is that the owner makes $120,000, and the other 4 employees each earn $30,000.Another possibility is that the owner makes $120,000, and the other 4 employees each earn $30,000.

Example 2, cont’d There are other possible situations, but these are enough to show that the salary the graduate could expect to earn can vary a lot based only on the mean salary.There are other possible situations, but these are enough to show that the salary the graduate could expect to earn can vary a lot based only on the mean salary.

The Median Median: the “middle number” of a data set when the values are arranged from smallest to largest.Median: the “middle number” of a data set when the values are arranged from smallest to largest. If there are an odd number of data points, the data point exactly in the middle is the median. If there are an odd number of data points, the data point exactly in the middle is the median. If there are an even number of data points, the mean of the two data points in the middle is the median. If there are an even number of data points, the mean of the two data points in the middle is the median.

Example 3 Find the mean and median of each data set.Find the mean and median of each data set. a) 0, 2, 4 b) 0, 2, 4, 10 c) 0, 2, 4, 10, 1000

Example 3 Find the mean and median of each data set.Find the mean and median of each data set. a) 0, 2, 4 mean: 2, median: 2 b) 0, 2, 4, 10 mean: 4, median: 3 c) 0, 2, 4, 10, 1000 mean: median: 4

Example 3, cont’d One very large or very small data value can change the mean dramatically.One very large or very small data value can change the mean dramatically. Large or small data values do not have much of an effect on the median.Large or small data values do not have much of an effect on the median.

Example 4 Find the median salary for the 2 situations.Find the median salary for the 2 situations. a) Five employees each earn $48,000. b) Four employees earn $30,000 and one earns $120,000.

Example 4, cont’d Solution:Solution: a) The median salary is $48,000. The median is the same as the mean.The median is the same as the mean. b) The median salary is $30,000. In this case the median more accurately shows the typical salary than the mean.In this case the median more accurately shows the typical salary than the mean.

Symmetric Distributions If the mean and median of a data set are equal, the data distribution is symmetric.If the mean and median of a data set are equal, the data distribution is symmetric. An example is shown below.An example is shown below.

Skewed Distributions Distribution is skewed left if the mean is less than the median.Distribution is skewed left if the mean is less than the median. Distribution is skewed right if the mean is more than the median.Distribution is skewed right if the mean is more than the median.

The Mode The mode is the most commonly- occurring value.The mode is the most commonly- occurring value. A data set may have:A data set may have: No mode. No mode. One mode. One mode. Multiple modes. Multiple modes.

Example 5 Find the mode(s) of the following test scores:Find the mode(s) of the following test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, 96.

Example 5 Find the mode(s) of the following test scores:Find the mode(s) of the following test scores: 26, 32, 54, 62, 67, 70, 71, 71, 74, 76, 80, 81, 84, 87, 87, 87, 89, 93, 95, appears 3 times, therefore it is the mode.

The Weighted Mean Weighted Mean: different data points have different levels of importance.Weighted Mean: different data points have different levels of importance. If the numbers,, have weightsIf the numbers,, have weights the weighted mean is:

Example 6 Suppose your grades are:Suppose your grades are: A in a 5-credit course A in a 5-credit course B in a 4-credit course B in a 4-credit course C in two 3-credit courses C in two 3-credit courses What is your GPA?What is your GPA?

Example 6, cont’d An A is worth 4 points, a B is 3 points, and a C is 2.An A is worth 4 points, a B is 3 points, and a C is 2. The weights are the number of credits.The weights are the number of credits. Your GPA is:Your GPA is:

Homework Pg. 573 Pg , 10, 14 2, 10, 14 Pg. 614 Pg , 4 2, 4