Excursions in Modern Mathematics Sixth Edition

1. Excursions in Modern MathematicsSixth Edition Peter Tannenbaum

2. Chapter 14Descriptive Statistics Graphing and Summarizing Data

3. Descriptive StatisticsOutline/learning Objectives • To interpret and produce an effective graphical summary of a data set. • To identify various types of numerical variables. • To interpret and produce numerical summaries of data including percentiles and five-number summaries.

4. Descriptive StatisticsOutline/learning Objectives • To describe the spread of a data set using range, interquartile range, and standard deviation.

5. Descriptive Statistics 14.1 Graphical Descriptions of Data

6. Descriptive Statistics • Data set A collection of data values denoted by N. • Data points Individual data values in a data set.

7. Descriptive Statistics Stat 101 Test Scores: Part 1 Professor Blackbeard has posted the results in the hallway outside his office. The data set consists of N = 75 data points (the number of students that took the test). Each data point is a raw score on the midterm between 0 and 25. Each student has one question on their mind: How did I do? It’s the next question that is statistically more interesting: How did the class as a whole do?

8. Descriptive Statistics Stat 101 Test Scores: Part 2 The first step in summarizing the information is to organize the scores in a frequency table. In this table, the number below each score gives the frequency of the score– that is, the number of students getting that particular score.

9. Descriptive Statistics Stat 101 Test Scores: Part 2 The figure below shows the information in a more visual way called a bar graph. With a bar graph, it is easy to detect outliers -- extreme data points that do not fit into the overall pattern of the data (the score of 1 and 24).

10. Descriptive Statistics Stat 101 Test Scores: Part 2 Sometimes it is more convenient to express the bar graph in a term of relative frequencies– that is, the frequencies given in terms of percentages of the total population.

11. Descriptive Statistics Stat 101 Test Scores: Part 2 Frequency charts that use icons or pictures instead of bars to show the frequencies are commonly referred to as pictograms.

12. Descriptive Statistics 14.2 Variables

13. Descriptive Statistics • Variable Any characteristic that varies with the members of a population. • Numerical (Quantitative) Variable A variable that represents a measurable quantity.

14. Descriptive Statistics • Continuous When the difference between the values of a numerical variable can be arbitrarily small. • Discrete When possible values of the numerical variable change by minimum increments.

15. Descriptive Statistics • Categorical (Qualitative) Variables Variables can also describe characteristics that cannot be measured numerically. • Pie Chart When the number of categories is small, another commonly used way to describe the relative frequencies of the categories.

16. Descriptive Statistics Stat 101 Test Scores: Part 3 The process of converting test scores (a numerical variable) into grades ( a categorical variable) requires setting up class intervals for the various letter grades. The grade distribution in the Stat 101 midterm can now be seen by means of a bar graph.

17. Descriptive Statistics Histograms When a numerical variable is continuous, its possible values can vary by infinitesimally small increments. As a consequence, there are no gaps between the class intervals.

18. Descriptive Statistics 14.3 Numerical Summaries of Data

19. Descriptive Statistics • Measures of Location The mean (or average), the median, and the quartiles are numbers that provide information about the values of the data. • Measures of Spread The range, the interquartile range, and the standard deviation are numbers that provide information about the spread within the data set.

20. Descriptive Statistics • Stat 101 Test Scores: Part 4 • The average of a set of N numbers is found by adding the numbers and dividing the total by N. • Step 1. Find the sum: Sum = d1•f1 + d2•f2 + … + dk•fk • = (1 • 1) + (6 • 1) +… + (24 • 1) = 814 • Step 2. Find N: N = f1 + f2 + … + fk = 75 • Step 3. Find A: A = Sum/N = 814/75  10.85

21. Descriptive Statistics • Percentile The pth percentile of a data set is a value such that p percent of the numbers fall at or below this value and the rest fall at or above it. • Locator Computed by the pth percent of N and is denoted by L. L = (p/100) •N

22. Descriptive Statistics Finding the pth Percentile of a Data Set • Step 0.Sort the data set. Let {d1, d2, d3, …, dN} represent the sorted data set. • Step 1. Find the locator: L = (p/100) •N • Step 2. Find the pth percentile: If L is a whole number, the pth percentile is given by d L.5. If L is not a whole number, the pth percentile is given by dL+ (L+ is L rounded up).

23. Descriptive Statistics The 50th percentile of a data set is known as the median and denoted by M. Finding the Median of a Data Set • Sort the data set. Let {d1, d2, d3, …, dN} represent the sorted data set. • If N is odd, the median is d(N+1)/2 . If N is even, the median is the average of dN/2 and d(N/2)+1 .

24. Descriptive Statistics After the median, the next most commonly used set of percentiles are the first and third quartiles. The first quartile (denoted by Q1) is the 25th percentile, and the third quartile (denoted by Q3) is the 75th percentile.

25. Descriptive Statistics Stat 101 Test Scores: Part 5 We will now find the median and quartile scored for Stat 101. Here N = 75 (odd), the median is d(75+1)/2= d38 . We conclude that the 38th test score is 11. Thus, M = 11. The locator for the first quartile is L = (0.25) X 75 = 18.75. We tally from left to right. Thus Q1 = d19 = 9. Sincethe first and third quartiles are at equal distance, a quick way to locate the third quartile is to count from right to left. Thus, Q3 = 12.

26. Descriptive Statistics A common way to summarize a large data set is by means of its five-number summary. The five-number summary is given by the smallest value in the data set (called the Min), the first quartile (Q1), the median (M), the third quartile (Q3), and the largest value in the data set (called the Max). These five numbers together often tells us a great deal about the data.

27. Descriptive Statistics Stat 101 Test Scores: Part 6 For the Stat 101 data set, the five-number summary is Min = 1, Q1 = 9, M = 11, Q3 = 12 and Max = 24. What useful information can we get out of this? The “big picture” we get from the five-number summary is that there were a lot of bunching up in a narrow band of scores. In general, this type of “lumpy” distribution of test scores is indicative of a test with an uneven level of difficulty.

28. Descriptive Statistics 14.4 Measures of Spread

29. Descriptive Statistics • Range The difference between the highest and lowest values of the data and is denoted by R. Thus, R = Max - Min. • Interquartile Range The difference between the third quartile and the first quartile (IQR = Q3 – Q1), and it tells us how spread out the middle 50% of the data values are.

30. Descriptive Statistics The Standard Deviation of a Data Set • Let A denote the mean of the data set. For each number x in the data set, compute its deviation from the mean (x – A), and square each of these numbers. These are called the squared deviations. • Find the average of the squared deviations. This number is called the variance V. • The standard deviation is the square root of the variance ( ).

31. Descriptive Statistics Conclusion • Basic concepts in statistics • Graphical summaries • Numerical summaries