STAT 101 Day 5 Descriptive Statistics II 13012

STAT 101: Day 5 Descriptive Statistics II 1/30/12 • One Quantitative Variable (continued) • Quantitative with a Categorical Variable • Two Quantitative Variables Section 2. 3, 2. 4, 2. 5 Professor Kari Lock Morgan Duke University

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Measures of Center m=$1, 250, 000 x=$2, 210, 000 Mean is “pulled” in the direction of skewness

Standard Deviation • The sample standard deviation, s, measures the spread of a distribution. The larger s is, the more spread out the distribution is Standard deviation is always ≥ 0. R: sd()

Standard Deviation Both of these distributions are bell-shaped

The 95% Rule • If a distribution is symmetric and bell-shaped, then approximately 95% of the data values will lie within 2 standard deviations of the mean

The 95% Rule The standard deviation for hours of sleep per night is closest to a) b) c) d) e) ½ 1 2 4 I have no idea

z-score • A z-score is unit-free measure of extremity of a data point. It tells us how many standard deviations away from the mean a value is • Values farther from 0 are more extreme • 95% of all z-scores fall between -2 and 2

z-score Which is better, an ACT score of 28 or a combined SAT score of 2100? • ACT: mean = 21, sd = 5 • SAT: mean = 1500, sd = 325 • Assume ACT scores and SAT scores have approximately symmetric and bell-shaped distributions (a) ACT score of 28 (b) SAT score of 2100 (c) I don’t know

Other Measures of Location Maximum = largest data value Minimum = smallest data value Quartiles: Q 1 = median of the values below m. Q 3 = median of the values above m.

Five Number Summary • Five Number Summary: Min Q 1 25% R: summary() 25% m Q 3 25% Max 25%

Percentile • The Pth percentile is the value of a quantitative variable which is greater than P percent of the data • We already used z-scores to determine whether an SAT score of 2100 or an ACT score of 28 is better • We could also have used percentiles: – ACT score of 28: 91 st percentile – SAT score of 2100: 97 th percentile

Five Number Summary • Five Number Summary: Min Q 1 25% 0 th percentile m 25% 25 th percentile Q 3 25% 50 th percentile Max 25% 75 th percentile 100 th percentile

Five Number Summary > summary(study_hours) Min. 1 st Qu. 2. 00 10. 00 Median 15. 00 3 rd Qu. 20. 00 Max. 69. 00 The distribution of number of hours you spend studying each week is (a) Symmetric (b) Right-skewed (c) Left-skewed (d) Impossible to tell

Measures of Spread • Range = Max – Min • Interquartile Range (IQR) = Q 3 – Q 1 • Is the range resistant to outliers? a) Yes b) No • Is the IQR resistant to outliers? a) Yes b) No

Outliers • Outliers can be informally identified by looking at a plot, but one rule of thumb for identifying outliers is data values more than 1. 5 IQRs beyond the quartiles • A data value is an outlier if it is Smaller than Q 1 – 1. 5(IQR) or Larger than Q 3 + 1. 5(IQR)

Boxplot Outliers • Lines (“whiskers”) extend from each quartile to the most extreme value that is not an outlier Q 3 Median Q 1 R: boxplot(study_hours, ylab=“Hours spent studying”)

Boxplot Which boxplot goes with the histogram of waiting times for the bus? (a) (b) (c)

Summary: One Quantitative Variable • Summary Statistics – – Center: mean, median Spread: standard deviation, range, IQR Percentiles 5 number summary • Visualization – Dotplot – Histogram – Boxplot • Other concepts – Shape: symmetric, skewed, bell-shaped – Outliers, resistance – z-scores

Quantitative and Categorical Relationships • Boxplots are particularly useful for comparing distributions of a quantitative variable across different levels of a categorical variable

Side-by-Side Boxplots Do students whose parents had more of an education have higher GPAs? boxplot(gpa~parent_degree, ylab="GPA", xlab="Parents' Highest Degree")

Side-by-Side Boxplots Does GPA differ by major?

Side-by-Side Boxplots Do students who’ve had AP statistics do better in STAT 101? NO!

Side-by-Side Boxplots

Quantitative Statistics by a Categorical Variable • Any of the statistics we use for a quantitative variable can be looked at separately for each level of a categorical variable • Mean hours per week spent studying by major:

Summary: One Quantitative and One Categorical • Summary Statistics – Any summary statistics for quantitative variables, broken down by each level of the categorical variable • Visualization – Side-by-side boxplots

Scatterplot • A scatterplot is a graph of the relationship between two quantitative variables. Each dot represents one case. R: plot(study_hours, gpa)

Direction of Association • A positive association means that values of one variable tend to be higher when values of the other variable are higher • A negative association means that values of one variable tend to be lower when values of the other variable are higher • Two variables are not associated if knowing the value of one variable does not give you any information about the value of the other variable

Cars Data - Handout • Quantitative Variables: – – – Weight (pounds) City MPG Fuel capacity (gallons) Page number (in Consumer Reports) Time to go ¼ mile (in seconds) Acceleration time from 0 to 60 mph • Relationships – – – Weight vs. City. MPG Weight vs. Fuel. Capacity Page. Num vs. Fuel Capacity Weight vs. Qtr. Mile Acc 060 vs. Qtr. Mile City. MPG vs. Qtr. Mile

Correlation • The sample correlation, r, measures the strength and direction of linear association between two quantitative variables s. X : sample standard deviation of X s. Y : sample standard deviation of Y R: cor(X, Y)

Car Correlations (-. 91) (. 89) (-. 45) (. 51) (-. 08) (. 99) What are the properties of correlation?

Correlation • -1 ≤ r ≤ 1 • positive association: r > 0 • negative association: r < 0 • no linear association: r 0 • The closer r is to ± 1, the stronger the linear association • r does not depend on the units of measurement • The correlation between X and Y is the same as the correlation between Y and X