Presentation is loading. Please wait.

Presentation is loading. Please wait.

Turning Data Into Information

Similar presentations


Presentation on theme: "Turning Data Into Information"— Presentation transcript:

1 Turning Data Into Information
Chapter 2 Turning Data Into Information Copyright ©2011 Brooks/Cole, Cengage Learning

2 2.1 Raw Data Raw data are for numbers and category labels that have been collected but have not yet been processed in any way. Example list of questions and raw data for a student: Copyright ©2011 Brooks/Cole, Cengage Learning

3 2.1 Raw Data An observation is an individual entity in a study.
A variable is a characteristic that may differ among individuals. Sample data are collected from a subset of a larger population. Population data are collected when all individuals in a population are measured. A statistic is a summary measure of sample data. A parameter is a summary measure of population data. Copyright ©2011 Brooks/Cole, Cengage Learning

4 2.2 Types of Variables Raw data from categorical variables consist of group or category names that don’t necessarily have a logical ordering. Examples: eye color, country of residence. Categorical variables for which the categories have a logical ordering are called ordinal variables. Examples: highest educational degree earned, tee shirt size (S, M, L, XL). Raw data from quantitative variables consist of numerical values taken on each individual. Examples: height, number of siblings. Copyright ©2011 Brooks/Cole, Cengage Learning

5 Asking the Right Questions
One Categorical Variable Example: What percentage of college students favor the legalization of marijuana, and what percentage of college students oppose legalization of marijuana? Ask: How many and what percentage of individuals fall into each category? Copyright ©2011 Brooks/Cole, Cengage Learning

6 Asking the Right Questions
Two Categorical Variables Example: In Case Study 1.6, we asked if the risk of having a heart attack was different for the physicians who took aspirin than for those who took a placebo. Ask: Is there a relationship between the two variables? Does the chance of falling into a particular category for one variable depend on which category an individual is in for the other variable? Copyright ©2011 Brooks/Cole, Cengage Learning

7 Asking the Right Questions
One Quantitative Variable Example: What is the average body temperature for adults, and how much variability is there in body temperature measurements? Ask: What are the interesting summary measures, like the average or the range of values? Copyright ©2011 Brooks/Cole, Cengage Learning

8 Asking the Right Questions
One Categorical and One Quantitative Variable Example: Do men and women drive at the same “fastest speeds” on average? Ask: Are the measurements similar across categories or do they differ? Could be asked regarding the averages or the ranges. Copyright ©2011 Brooks/Cole, Cengage Learning

9 Asking the Right Questions
Two Quantitative Variables Example: Does average body temperature change as people age? Ask: Are these variables related so that when measurements are high (or low) on one variable the measurements for the other variable also tend to be high (or low)? Copyright ©2011 Brooks/Cole, Cengage Learning

10 Explanatory and Response Variables
Many questions about the relationship between two variables. It is useful to identify one variable as the explanatory variable and the other variable as the response variable. In general, the value of the explanatory variable for an individual is thought to partially explain the value of the response variable for that individual. Copyright ©2011 Brooks/Cole, Cengage Learning

11 2.3 Summarizing One or Two Categorical Variables
Numerical Summaries Count how many fall into each category. Calculate the percent in each category. If two variables, have the categories of the explanatory variable define the rows and compute row percentages. Copyright ©2011 Brooks/Cole, Cengage Learning

12 Example 2.1 Seatbelt Use by Twelfth-Graders
2003 nationwide survey of American HS students “How often do you wear a seatbelt when driving a car?” Total sample size n = 3042 students. A majority, 1686/3042 = .554, or 55.4%, said they always wear a seatbelt, while 115/3042 = .038, or 3.8%, said they never wear a seatbelt. Rarely or never: 8.2% + 3.8% =12% Copyright ©2011 Brooks/Cole, Cengage Learning

13 Example 2.1 Seatbelt Use by Twelfth-Graders
Are females more likely to say always wear seatbelt? Are males more likely to say rarely or never wear seatbelt? Females: 915/1467 = 62.4% said always wear seatbelt Males: 771/1575 = 49.0% said always wear seatbelt. Males: 10.5% + 5.7% =16.2% rarely or never wear one. Females: 5.7% + 1.7% = 7.4% rarely or never wear one. Copyright ©2011 Brooks/Cole, Cengage Learning

14 2.3 Summarizing One or Two Categorical Variables
Frequency and Relative Frequency A frequency distribution for a categorical variable is a listing of all categories along with their frequencies (counts). A relative frequency distribution is a listing of all categories along with their relative frequencies (given as proportions or percentages, for example). Copyright ©2011 Brooks/Cole, Cengage Learning

15 Example 2.2 Lighting the Way to Nearsightedness
Survey of n = 479 children. Those who slept with nightlight or in fully lit room before age 2 had higher incidence of nearsightedness (myopia) later in childhood. Note: Study does not prove sleeping with light actually caused myopia in more children. Copyright ©2011 Brooks/Cole, Cengage Learning

16 Visual Summaries for Categorical Variables
Pie Charts: useful for summarizing a single categorical variable if not too many categories. Bar Graphs: useful for summarizing one or two categorical variables and particularly useful for making comparisons when there are two categorical variables. Copyright ©2011 Brooks/Cole, Cengage Learning

17 Example 2.3 Humans Are Not Good Randomizers
Survey of n = 190 college students “Randomly pick a number between 1 and 10.” Results: Most chose 7, very few chose 1 or 10. Copyright ©2011 Brooks/Cole, Cengage Learning

18 Example 2.4 Revisiting Nightlights and Nearsightedness
Survey of n = 479 children. Response: Degree of Myopia Explanatory: Amount of Sleeptime Lighting Copyright ©2011 Brooks/Cole, Cengage Learning

19 2.4 Finding Information in Quantitative Data
Long list of numbers – needs to be organized to obtain answers to questions of interest. Copyright ©2011 Brooks/Cole, Cengage Learning

20 Five-Number Summaries
Find extremes (high, low), the median, and the quartiles (medians of lower and upper halves of the values). Quick overview of the data values. Information about the center, spread, and shape of data. Copyright ©2011 Brooks/Cole, Cengage Learning

21 Example 2.5 Right Handspans
Majority of females had handspans between 19 and 21 cm, and many males had handspans between 21.5 and 23 cm. Two females with unusually small handspans. Copyright ©2011 Brooks/Cole, Cengage Learning

22 Example 2.5 Right Handspans
About 25% of handspans of females are between 12.5 and 19.0 centimeters, about 25% are between 19 and 20 cm, about 25% are between 20 and 21 cm, and about 25% are between 21 and cm. Copyright ©2011 Brooks/Cole, Cengage Learning

23 Summary Features of Quantitative Variables
Location: center or average. e.g. median Spread: variability e.g. difference between two extremes or two quartiles. Shape: clumped in middle or on one end (more later) Outliers: a data point that is not consistent with the bulk of the data Copyright ©2011 Brooks/Cole, Cengage Learning

24 Example 2.6 Annual Compensation for Highest Paid CEOs in the United States
Paid compensation (in millions of $) for 50 highest-paid CEOs in 2008 for Fortune Magazine’s Top 500 companies Median: ~ $35.6 million Minimum: $24.3 million Maximum: $557 million (perhaps outlier?) Shape: most clumped on lower end (= skewed) Copyright ©2011 Brooks/Cole, Cengage Learning

25 Example 2.7 Ages of Death of U.S. First Ladies
Partial Data Listing and five-number summary: Extremes are more interesting here: Who died at 34? Martha Jefferson Who lived to be 97? Bess Truman Copyright ©2011 Brooks/Cole, Cengage Learning

26 Pictures for Quantitative Data
Histograms: similar to bar graphs, used for any number of data values. Stem-and-leaf plots and dotplots: present all individual values, useful for small to moderate sized data sets. Boxplot or box-and-whisker plot: useful summary for comparing two or more groups. Copyright ©2011 Brooks/Cole, Cengage Learning

27 Interpreting Histograms, Stemplots, and Dotplots
Values are centered around 20 cm. Two possible low outliers. Apart from outliers, spans range from about 16 to 23 cm. Copyright ©2011 Brooks/Cole, Cengage Learning

28 Creating a Histogram Step 1: Decide how many equally spaced (same width) intervals to use for the horizontal axis. Between 6 and 15 intervals is a good number. Step 2: Decide to use frequencies (count) or relative frequencies (proportion) on the vertical axis. Step 3: Draw equally spaced intervals on horizontal axis covering entire range of the data. Determine frequency or relative frequency of data values in each interval and draw a bar with corresponding height. Decide rule to use for values that fall on the border between two intervals. Copyright ©2011 Brooks/Cole, Cengage Learning

29 Example 2.8 Ages of Death of First Ladies
Two different histograms Copyright ©2011 Brooks/Cole, Cengage Learning

30 Creating a Dotplot Draw a number line (horizontal axis) to cover range from smallest to largest data value. For each observation, place a dot above the number line located at the observation’s data value. When multiple observations with the same value, dots are stacked vertically Copyright ©2011 Brooks/Cole, Cengage Learning

31 Creating a Stem-and-Leaf Plot
Step 1: Determine stem values. The “stem” contains all but the last of the displayed digits of a number. Stems should define equally spaced intervals. Step 2: For each individual, attach a “leaf” to the appropriate stem. A “leaf” is the last of the displayed digits of a number. Often leaves are ordered on each stem. Note: More than one way to define stems. Can use split-stems or truncate/round values first. Copyright ©2011 Brooks/Cole, Cengage Learning

32 Example 2.9 Big Music Collection
How many songs on iPod or MP3? 2510, 500, 500, 1300, 687, 600, 500, 2600, 30, 900, 800, 0, 750, 1500, 1500, 2400, 800, 2017, 1150, 5000, 4000, 1250, 1700, 3305 Final two digits truncated 2510: stem label of 2 and leaf value of 5 500: stem label of 0 and leaf value of 5 30: stem value is 0 and leaf value also 0 Two stems for each 1000s possibility: first = leaf values 0, 1, 2, 3, 4 second = leaf values 5, 6, 7, 8, 9 Shape is skewed right Copyright ©2011 Brooks/Cole, Cengage Learning

33 Describing Shape Symmetric, bell-shaped Symmetric, not bell-shaped
Skewed Right: values trail off to right Skewed Left: values trail off to left Copyright ©2011 Brooks/Cole, Cengage Learning

34 Boxplots: Picturing Location and Spread for Group Comparisons
Box covers the middle 50% of the data Line within box marks the median value Possible outliers are marked with asterisk Apart from outliers, lines extending from box reach to min and max values. Copyright ©2011 Brooks/Cole, Cengage Learning

35 Using Visual Displays To illustrate location and spread, any of the pictures work well. To illustrate shape, histograms and stem-and-leaf plots are best. To see individual values, use stem-and-leaf plots and dotplots. To sort values, use stem-and-leaf plots. To compare groups, use side-by-side boxplots. To identify outliers using the standard definition, use a boxplot. Copyright ©2011 Brooks/Cole, Cengage Learning

36 2.6 Numerical Summaries of Quantitative Data
Notation for Raw Data: n = number of individuals in a data set x1, x2 , x3,…, xn represent individual raw data values Example: A data set consists of handspan values in centimeters for six females; the values are 21, 19, 20, 20, 22, and Then, n = 6 x1= 21, x2 = 19, x3 = 20, x4 = 20, x5 = 22, and x6 = 19 Copyright ©2011 Brooks/Cole, Cengage Learning

37 Describing the Location of a Data Set
Mean: the numerical average Median: the middle value (if n odd) or the average of the middle two values (n even) Symmetric: mean = median Skewed Left: mean < median Skewed Right: mean > median Copyright ©2011 Brooks/Cole, Cengage Learning

38 Determining the Mean and Median
where means “add together all the values” The Median If n is odd: M = middle of ordered values. Count (n + 1)/2 down from top of ordered list. If n is even: M = average of middle two ordered values. Average values that are (n/2) and (n/2) + 1 down from top of ordered list. Copyright ©2011 Brooks/Cole, Cengage Learning

39 Example 2.12 Will “Normal” Rainfall Get Rid of Those Odors?
Data: Average rainfall (inches) for Davis, California for 47 years Mean = inches Median = inches In , a company with odor problem blamed it on excessive rain. That year rainfall was inches. More rain occurred in 4 other years. Copyright ©2011 Brooks/Cole, Cengage Learning

40 The Influence of Outliers on the Mean and Median
Larger influence on mean than median. High outliers will increase the mean. Low outliers will decrease the mean. If ages at death are: 76, 78, 80, 82, and 84 then mean = median = 80 years. If ages at death are: 46, 78, 80, 82, and 84 then median = 80 but mean = 74 years. Copyright ©2011 Brooks/Cole, Cengage Learning

41 Describing Spread: Range and Interquartile Range
Range = high value – low value Interquartile Range (IQR) = upper quartile – lower quartile Standard Deviation (covered later in Section 2.7) Copyright ©2011 Brooks/Cole, Cengage Learning

42 Example 2.13 Fastest Speeds Ever Driven
Five-Number Summary for 87 males Median = 110 mph measures the center of the data Two extremes describe spread over 100% of data Range = 150 – 55 = 95 mph Two quartiles describe spread over middle 50% of data Interquartile Range = 120 – 95 = 25 mph Copyright ©2011 Brooks/Cole, Cengage Learning

43 Notation and Finding the Quartiles
Split the ordered values into the half that is below the median and the half that is above the median. Q1 = lower quartile = median of data values that are below the median Q3 = upper quartile = median of data values that are above the median Copyright ©2011 Brooks/Cole, Cengage Learning

44 Example 2.14 Fastest Speeds for Men
Ordered Data (in rows of 10 values) for the 87 males: Median = (87+1)/2 = 44th value in the list = 110 mph Q1 = median of the 43 values below the median = (43+1)/2 = 22nd value from the start of the list = 95 mph Q3 = median of the 43 values above the median = (43+1)/2 = 22nd value from the end of the list = 120 mph Copyright ©2011 Brooks/Cole, Cengage Learning

45 How to Draw a Boxplot and Identify Outliers
Step 1: Label either a vertical axis or a horizontal axis with numbers from min to max of the data. Step 2: Draw box with lower end at Q1 and upper end at Q3. Step 3: Draw a line through the box at the median M. Step 4: Calculate IQR = Q3 – Q1. Step 5: Draw a line from Q1 end of box to smallest data value that is not further than 1.5  IQR from Q1. Draw a line from Q3 end of box to largest data value that is not further than 1.5  IQR from Q3. Step 6: Mark data points further than 1.5  IQR from either edge of the box with an asterisk. Points represented with asterisks are considered to be outliers. Copyright ©2011 Brooks/Cole, Cengage Learning

46 Percentiles The kth percentile is a number that has k% of the data values at or below it and (100 – k)% of the data values at or above it. Lower quartile = 25th percentile Median = 50th percentile Upper quartile = 75th percentile Copyright ©2011 Brooks/Cole, Cengage Learning

47 2.6 How to Handle Outliers Outlier: a data point that is not consistent with the bulk of the data. Look for them via graphs. Can have big influence on conclusions. Can cause complications in some statistical analyses. Cannot discard without justification. Copyright ©2011 Brooks/Cole, Cengage Learning

48 Possible Reasons for Outliers and Reasonable Actions
Outlier is legitimate data value and represents natural variability for the group and variable(s) measured. Values may not be discarded — they provide important information about location and spread. Mistake made while taking measurement or entering it into computer. If verified, should be discarded/corrected. Individual in question belongs to a different group than bulk of individuals measured. Values may be discarded if summary is desired and reported for the majority group only. Copyright ©2011 Brooks/Cole, Cengage Learning

49 Example 2.16 Tiny Boatsmen Weights (in pounds) of 18 men on crew team:
Cambridge:188.5, 183.0, 194.5, 185.0, 214.0, , 186.0, 178.5, 109.0 Oxford: , 184.5, 204.0, 184.5, 195.5, , 174.0, 183.0, 109.5 Note: last weight in each list is unusually small. They are the coxswains for their teams, while others are rowers. Copyright ©2011 Brooks/Cole, Cengage Learning

50 2.7 Bell-Shaped Distributions of Numbers
Many measurements follow a predictable pattern: Most individuals are clumped around the center The greater the distance a value is from the center, the fewer individuals have that value. Variables that follow such a pattern are said to be “bell-shaped”. A special case is called a normal distribution or normal curve. Copyright ©2011 Brooks/Cole, Cengage Learning

51 Example 2.17 Bell-Shaped British Women’s Heights
Data: representative sample of 199 married British couples. Below shows a histogram of the wives’ heights with a normal curve superimposed. The mean height = 1602 millimeters. Copyright ©2011 Brooks/Cole, Cengage Learning

52 Describing Spread with Standard Deviation
Standard deviation measures variability by summarizing how far individual data values are from the mean. Think of the standard deviation as roughly the average distance values fall from the mean. Copyright ©2011 Brooks/Cole, Cengage Learning

53 Describing Spread with Standard Deviation
Both sets have same mean of 100. Set 1: all values are equal to the mean so there is no variability at all. Set 2: one value equals the mean and other four values are 10 points away from the mean, so the average distance away from the mean is about 10. Copyright ©2011 Brooks/Cole, Cengage Learning

54 Calculating the Standard Deviation
Formula for the (sample) standard deviation: The value of s2 is called the (sample) variance. An equivalent formula, easier to compute, is: Copyright ©2011 Brooks/Cole, Cengage Learning

55 Calculating the Standard Deviation
Step 1: Calculate , the sample mean. Step 2: For each observation, calculate the difference between the data value and the mean. Step 3: Square each difference in step 2. Step 4: Sum the squared differences in step 3, and then divide this sum by n – 1. Step 5: Take the square root of the value in step 4. Copyright ©2011 Brooks/Cole, Cengage Learning

56 Example 2.18 Calculating a Standard Deviation
Consider four pulse rates: 62, 68, 74, 76 Step 1: Steps 2 and 3: Step 4: Step 5: Copyright ©2011 Brooks/Cole, Cengage Learning

57 Population Mean and Standard Deviation
Data sets usually represent a sample from a larger population. If the data set includes measurements for an entire population, the notations for the mean and standard deviation are different, and the formula for the standard deviation is also slightly different. A population mean is represented by the symbol m (“mu”), and the population standard deviation is Copyright ©2011 Brooks/Cole, Cengage Learning

58 Interpreting the Standard Deviation for Bell-Shaped Curves:
Interpreting the Standard Deviation for Bell-Shaped Curves: The Empirical Rule For any bell-shaped curve, approximately 68% of the values fall within 1 standard deviation of the mean in either direction 95% of the values fall within 2 standard deviations of the mean in either direction 99.7% of the values fall within 3 standard deviations of the mean in either direction Note: ~0.3% fall farther than 3 standard deviations from mean Copyright ©2011 Brooks/Cole, Cengage Learning

59 Example 2.19 Women’s Heights revisited
Mean height for the 199 British women is 1602 mm and standard deviation is 62.4 mm. 68% of the 199 heights would fall in the range 1602  62.4, or to mm 95% of the heights would fall in the interval  2(62.4), or to mm 99.7% of the heights would fall in the interval 1602  3(62.4), or to mm Copyright ©2011 Brooks/Cole, Cengage Learning

60 Example 2.19 Women’s Heights revisited
Note: Not perfect, but follows Empirical Rule quite well Copyright ©2011 Brooks/Cole, Cengage Learning

61 The Empirical Rule, the Standard Deviation, and the Range
Empirical Rule => the range from the minimum to the maximum data values equals about 4 to 6 standard deviations for data with an approximate bell shape. You can get a rough idea of the value of the standard deviation by dividing the range by 6. Copyright ©2011 Brooks/Cole, Cengage Learning

62 Standardized z-Scores
Standardized score or z-score: Example: Mean resting pulse rate for adult men is 70 beats per minute (bpm), standard deviation is 8 bpm. The standardized score for a resting pulse rate of 80: A pulse rate of 80 is 1.25 standard deviations above the mean pulse rate for adult men. Copyright ©2011 Brooks/Cole, Cengage Learning

63 The Empirical Rule Restated
For bell-shaped data, About 68% of values have z-scores between –1 and +1. About 95% of values have z-scores between –2 and +2. About 99.7% of values have z-scores between –3 and +3. Copyright ©2011 Brooks/Cole, Cengage Learning


Download ppt "Turning Data Into Information"

Similar presentations


Ads by Google