1. What is Statistics?
Statistics The science of collecting, organizing, analyzing, and interpreting data in order to make decisions.
Larson/Farber 4th ed.

2. What is Data?
Data Consist of information coming from observations, counts, measurements, or responses.
Larson/Farber 4th ed.

3. Data Sets
Population
The collection of all outcomes, responses, measurements, or counts that are of interest.
Sample
A subset of the population.
Larson/Farber 4th ed.

4. The entire group of individuals to be studied is called the population
The entire group of individuals to be studied is called the population. An individual is a person or object that is a member of the population being studied. A sample is a subset of the population that is being studied.

5. Parameter and Statistic
A number that describes a population characteristic.
Average age of all people in the United States
Statistic
A number that describes a sample characteristic.
Average age of people from a sample of three states
Larson/Farber 4th ed.

6. Branches of Statistics
Descriptive Statistics Involves organizing, summarizing, and displaying data.
e.g. Tables, charts, averages
Inferential Statistics Involves using sample data to draw conclusions about a population.
Larson/Farber 4th ed.

7. Section 1.2
Data Classification
Larson/Farber 4th ed.

8. Qualitative or Categorical variables allow for classification of individuals based on some attribute or characteristic.
Quantitative variables provide numerical measures of individuals. Arithmetic operations such as addition and subtraction can be performed on the values of the quantitative variable and provide meaningful results.

9. c. Household income in the previous year d. Level of education
Researcher Elisabeth Kvaavik and others studied factors that affect the eating habits of adults in their mid-thirties. (Source: Kvaavik E, et. al. Psychological explanatorys of eating habits among adults in their mid-30’s (2005) International Journal of Behavioral Nutrition and Physical Activity (2)9.) Classify each of the following variables considered in the study as qualitative or quantitative.
a. Nationality
Number of children
c. Household income in the previous year
d. Level of education
e. Daily intake of whole grains (measured in grams per day)
Qualitative
Quantitative
Quantitative
Qualitative
Quantitative

10. Types of Data
Qualitative Data Consists of attributes, labels, or nonnumerical entries.
Major
Place of birth
Eye color
Larson/Farber 4th ed.

11. Types of Data Quantitative data Numerical measurements or counts. Age
Weight of a letter
Temperature
Larson/Farber 4th ed.

12. Solution: Classifying Data by Type
Qualitative Data (Names of vehicle models are nonnumerical entries)
Quantitative Data (Base prices of vehicles models are numerical entries)
Larson/Farber 4th ed.

13. A discrete variable is a quantitative variable that either has a finite number of possible values or a countable number of possible values. The term “countable” means the values result from counting such as 0, 1, 2, 3, and so on.
A continuous variable is a quantitative variable that has an infinite number of possible values it can take on and can be measured to any desired level of accuracy.

14. Sampling Techniques Simple Random Sample
Every possible sample of the same size has the same chance of being selected. x
Larson/Farber 4th ed.

15. Simple Random Sample
Random numbers can be generated by a random number table, a software program or a calculator.
Assign a number to each member of the population.
Members of the population that correspond to these numbers become members of the sample.
Larson/Farber 4th ed.

16. Solution: Simple Random Sample
Read the digits in groups of three
Ignore numbers greater than 731
The students assigned numbers 719, 662, 650, 4, 53, 589, 403, and 129 would make up the sample.
Larson/Farber 4th ed.

17. Other Sampling Techniques
Systematic Sample
Choose a starting value at random. Then choose every kth member of the population.
In the West Ridge County example you could assign a different number to each household, randomly choose a starting number, then select every 100th household.
Larson/Farber 4th ed.

18. Other Sampling Techniques
Stratified Sample
Divide a population into groups (strata) and select a random sample from each group.
To collect a stratified sample of the number of people who live in West Ridge County households, you could divide the households into socioeconomic levels and then randomly select households from each level.
Larson/Farber 4th ed.

19. Other Sampling Techniques
Cluster Sample
Divide the population into groups (clusters) and select all of the members in one or more, but not all, of the clusters.
In the West Ridge County example you could divide the households into clusters according to zip codes, then select all the households in one or more, but not all, zip codes.
Larson/Farber 4th ed.

20. A convenience sample is one in which the individuals in the sample are easily obtained.
Any studies that use this type of sampling generally have results that are suspect. Results should be looked upon with extreme skepticism.

21. Example: Identifying Sampling Techniques
You are doing a study to determine the opinion of students at your school regarding stem cell research. Identify the sampling technique used.
You divide the student population with respect to majors and randomly select and question some students in each major.
Solution:
Stratified sampling (the students are divided into strata (majors) and a sample is selected from each major)
Larson/Farber 4th ed.

22. Example: Identifying Sampling Techniques
You assign each student a number and generate random numbers. You then question each student whose number is randomly selected.
Solution:
Simple random sample (each sample of the same size has an equal chance of being selected and each student has an equal chance of being selected.)
Larson/Farber 4th ed.

23. Descriptive Statistics
Measures of Central Tendency
Larson/Farber 4th ed.

24. Measures of Central Tendency
Measure of central tendency
A value that represents a typical, or central, entry of a data set.
Most common measures of central tendency:
Mean
Median
Mode
Larson/Farber 4th ed.

25. Measure of Central Tendency: Mean
Mean (average)
The sum of all the data entries divided by the number of entries.
Sigma notation: Σx = add all of the data entries (x) in the data set.
Population mean:
Sample mean:
Larson/Farber 4th ed.

26. Example: Finding a Sample Mean
The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. What is the mean price of the flights?
Larson/Farber 4th ed.

27. Solution: Finding a Sample Mean
The sum of the flight prices is
Σx = = 3695
To find the mean price, divide the sum of the prices by the number of prices in the sample
The mean price of the flights is about $
Larson/Farber 4th ed.

28. Measure of Central Tendency: Median
The value that lies in the middle of the data when the data set is ordered.
Measures the center of an ordered data set by dividing it into two equal parts.
If the data set has an
odd number of entries: median is the middle data entry.
even number of entries: median is the mean of the two middle data entries.
Larson/Farber 4th ed.

29. Example: Finding the Median
The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the median of the flight prices
Larson/Farber 4th ed.

30. Solution: Finding the Median
First order the data.
There are seven entries (an odd number), the median is the middle, or fourth, data entry.
The median price of the flights is $427.
Larson/Farber 4th ed.

31. Example: Finding the Median
The flight priced at $432 is no longer available. What is the median price of the remaining flights?
Larson/Farber 4th ed.

32. Solution: Finding the Median
First order the data.
There are six entries (an even number), the median is the mean of the two middle entries.
The median price of the flights is $412.
Larson/Farber 4th ed.

33. Measure of Central Tendency: Mode
The data entry that occurs with the greatest frequency.
If no entry is repeated the data set has no mode.
If two entries occur with the same greatest frequency, each entry is a mode (bimodal).
Larson/Farber 4th ed.

34. Example: Finding the Mode
The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the mode of the flight prices
Larson/Farber 4th ed.

35. Solution: Finding the Mode
Ordering the data helps to find the mode.
The entry of 397 occurs twice, whereas the other data entries occur only once.
The mode of the flight prices is $397.
Larson/Farber 4th ed.

36. Example: Finding the Mode
At a political debate a sample of audience members was asked to name the political party to which they belong. Their responses are shown in the table. What is the mode of the responses?
Political Party
Frequency, f
Democrat 34
Republican 56
Other 21
Did not respond 9
Larson/Farber 4th ed.

37. Solution: Finding the Mode
Political Party
Frequency, f
Democrat 34
Republican 56
Other 21
Did not respond 9
The mode is Republican (the response occurring with the greatest frequency). In this sample there were more Republicans than people of any other single affiliation.
Larson/Farber 4th ed.

38. Comparing the Mean, Median, and Mode
All three measures describe a typical entry of a data set.
Advantage of using the mean:
The mean is a reliable measure because it takes into account every entry of a data set.
Disadvantage of using the mean:
Greatly affected by outliers (a data entry that is far removed from the other entries in the data set).
Larson/Farber 4th ed.

39. Example: Comparing the Mean, Median, and Mode
Find the mean, median, and mode of the sample ages of a class shown. Which measure of central tendency best describes a typical entry of this data set? Are there any outliers?
Ages in a class 20 21 22 23 24 65
Larson/Farber 4th ed.

40. Solution: Comparing the Mean, Median, and Mode
Ages in a class 20 21 22 23 24 65
Mean:
Median:
Mode:
20 years (the entry occurring with the greatest frequency)
Larson/Farber 4th ed.

41. Solution: Comparing the Mean, Median, and Mode
Mean ≈ 23.8 years Median = 21.5 years Mode = 20 years
The mean takes every entry into account, but is influenced by the outlier of 65.
The median also takes every entry into account, and it is not affected by the outlier.
In this case the mode exists, but it doesn't appear to represent a typical entry.
Larson/Farber 4th ed.

42. Mean of Grouped Data Mean of a Frequency Distribution Approximated by
where x and f are the midpoints and frequencies of a class, respectively
Larson/Farber 4th ed.

43. Section 2.3 Summary
Determined the mean, median, and mode of a population and of a sample
Determined the weighted mean of a data set and the mean of a frequency distribution
Described the shape of a distribution as symmetric, uniform, or skewed and compared the mean and median for each
Larson/Farber 4th ed.

44. Range
Range
The difference between the maximum and minimum data entries in the set.
The data must be quantitative.
Range = (Max. data entry) – (Min. data entry)
Larson/Farber 4th ed.

45. Example: Finding the Range
A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the range of the starting salaries. Starting salaries (1000s of dollars)
Larson/Farber 4th ed.

46. Solution: Finding the Range
Ordering the data helps to find the least and greatest salaries.
Range = (Max. salary) – (Min. salary)
= 47 – 37 = 10
The range of starting salaries is 10 or $10,000.
minimum
maximum
Larson/Farber 4th ed.

47. Section 2.5
Measures of Position
Larson/Farber 4th ed.

48. Section 2.5 Objectives Determine the quartiles of a data set
Determine the interquartile range of a data set
Create a box-and-whisker plot
Interpret other fractiles such as percentiles
Determine and interpret the standard score (z-score)
Larson/Farber 4th ed.

49. Quartiles
Fractiles are numbers that partition (divide) an ordered data set into equal parts.
Quartiles approximately divide an ordered data set into four equal parts.
First quartile, Q1: About one quarter of the data fall on or below Q1.
Second quartile, Q2: About one half of the data fall on or below Q2 (median).
Third quartile, Q3: About three quarters of the data fall on or below Q3.
Larson/Farber 4th ed.

50. Example: Finding Quartiles
The test scores of 15 employees enrolled in a CPR training course are listed. Find the first, second, and third quartiles of the test scores
Solution:
Q2 divides the data set into two halves.
Lower half
Upper half Q2
Larson/Farber 4th ed.

51. Solution: Finding Quartiles
The first and third quartiles are the medians of the lower and upper halves of the data set.
Lower half
Upper half Q1 Q2 Q3
About one fourth of the employees scored 10 or less, about one half scored 15 or less; and about three fourths scored 18 or less.
Larson/Farber 4th ed.

52. Interquartile Range Interquartile Range (IQR)
The difference between the third and first quartiles.
IQR = Q3 – Q1
Larson/Farber 4th ed.

53. Example: Finding the Interquartile Range
Find the interquartile range of the test scores. Recall Q1 = 10, Q2 = 15, and Q3 = 18
Solution:
IQR = Q3 – Q1 = 18 – 10 = 8
The test scores in the middle portion of the data set vary by at most 8 points.
Larson/Farber 4th ed.

54. Box-and-Whisker Plot Box-and-whisker plot
Exploratory data analysis tool.
Highlights important features of a data set.
Requires (five-number summary):
Minimum entry
First quartile Q1
Median Q2
Third quartile Q3
Maximum entry
Larson/Farber 4th ed.

55. Drawing a Box-and-Whisker Plot
Find the five-number summary of the data set.
Construct a horizontal scale that spans the range of the data.
Plot the five numbers above the horizontal scale.
Draw a box above the horizontal scale from Q1 to Q3 and draw a vertical line in the box at Q2.
Draw whiskers from the box to the minimum and maximum entries.
Whisker
Maximum entry
Minimum entry
Box
Median, Q2 Q3 Q1
Larson/Farber 4th ed.

56. Example: Drawing a Box-and-Whisker Plot
Draw a box-and-whisker plot that represents the 15 test scores. Recall Min = 5 Q1 = 10 Q2 = 15 Q3 = 18 Max = 37
Solution: 5 10 15 18 37
About half the scores are between 10 and 18. By looking at the length of the right whisker, you can conclude 37 is a possible outlier.
Larson/Farber 4th ed.

57. The Shape of Distributions
Symmetric Distribution
A vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images.
Larson/Farber 4th ed.

58. The Shape of Distributions
Uniform Distribution (rectangular)
All entries or classes in the distribution have equal or approximately equal frequencies.
Symmetric.
Larson/Farber 4th ed.

59. The Shape of Distributions
Skewed Left Distribution (negatively skewed)
The “tail” of the graph elongates more to the left.
The mean is to the left of the median.
Larson/Farber 4th ed.

60. The Shape of Distributions
Skewed Right Distribution (positively skewed)
The “tail” of the graph elongates more to the right.
The mean is to the right of the median.
Larson/Farber 4th ed.

61. © 2010 Pearson Prentice Hall. All rights reserved
The interest rate boxplot indicates that the distribution is skewed left.
© 2010 Pearson Prentice Hall. All rights reserved
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62. © 2010 Pearson Prentice Hall. All rights reserved
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