Describing Data: Two Variables

Transcription

1 STAT 250 Dr. Kari Lock Morgan Describing Data: Two Variables SECTIONS 2.4, 2.5 One quantitative variable (2.4) One quantitative and one categorical (2.4) Two quantitative (2.5) z- score Which is better, an ACT score of 28 or a combined SAT score of 2100? ACT: μ = 21, σ = 5 SAT: μ = 1500, σ = 325 Assume ACT and SAT scores have approximately 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 m Q 3 25% 25% 25% 25% Max Minitab: Stat -> Basic Statistics -> Display Descriptive Statistics Five Number Summary Percentile > summary(study_hours) Min. 1st Qu. Median 3rd Qu. Max The distribution of number of hours spent studying each week is a) Symmetric b) Right- skewed c) Left- skewed d) Impossible to tell The P th is the value which is greater than P% 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 s: ACT score of 28: 91st SAT score of 2100: 97th 1

2 Five Number Summary Five Number Summary: Measures of Spread Range = Max Min Min Q 1 m Q 3 25% 25% 25% 25% Max Interquartile Range (IQR) = Q 3 Q 1 Is the range resistant to outliers? 0 th 25 th 50 th 75 th 100 th Is the IQR resistant to outliers? Measures of Center: Mean (not resistant) Median (resistant) Comparing Statistics Measures of Spread: Standard deviation (not resistant) IQR (resistant) Range (not resistant) Most often, we use the mean and the standard deviation, because they are calculated based on all the data values, so use all the available information Middle 50% of data Boxplot Lines ( whiskers ) extend from each quartile to the most extreme value that is not an outlier Q 3 Q 1 Median Minitab: Graph -> Boxplot -> One Y -> Simple Boxplot Boxplot *For boxplots, outliers are deeined as any point more than 1.5 IQRs beyond the quartiles (although you don t have to know that) Outlier This boxplot shows a distribution that is a) Symmetric b) Left- skewed c) Right- skewed 2

3 Summary: One Quantitative Variable Summary Statistics Center: mean, median Spread: standard deviation, range, IQR 5 number summary Percentiles Honeybee Waggle Dances Visualization Dotplot Histogram Boxplot Other concepts Shape: symmetric, skewed, bell- shaped Outliers, resistance z- scores 7ijI- g4jhg Honeybee Waggle Dance Honeybee scouts investigate new home or food source options; the scouts communicate the information to the hive with a waggle dance Scientists took bees to an island with only two possible options for nesting: one of very high quality and one of low quality. They recorded Quality of nesting site Distance to nesting site Number of waggle dance circuits performed Duration of waggle dance Seeley, T., Honeybee Democracy, Princeton University Press, Princeton, NJ, 2010, p. 128 Questions of the Day How many circuits of the waggle dance do honey bees do? How is this related to quality of a nesting site? How is duration of the dance related to distance to a nesting site? Review: One Quantitative How many circuits of the waggle dance do bees do? (One quantitative variable) One Quantitative and One Categorical How is number of waggle circuits related to the quality of the nesting site? Two variables One quantitative (number of circuits) One categorical (quality low or high) Can do anything for one quantitative variable, broken down by categorical groups 3

4 Side- by- Side Boxplots Stacked Dotplots Minitab: Graph -> Boxplot -> One Y -> With Groups Minitab: Graph -> Dotplot -> One Y -> With Groups Overlaid Histograms 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 Minitab: Graph -> Histogram -> With Groups Minitab: Stat -> Basic Statistics -> Display Descriptive Statistics -> By variables Difference in Means Often, when comparing a quantitative variable across two categories, we compute the difference in means Association? Does there appear to be an association between number of waggle circuits and quality of potential nesting site?! x x = = 60.5 H L Honeybees perform 60.5 circuits more, on average, for the high quality site as opposed to the low quality site. 4

5 Summary: One Quantitative and One Categorical Summary Statistics Any summary statistics for quantitative variables, broken down by groups Difference in means Visualization Side- by- side graphs Two Quantitative Variables How is duration of the dance related to distance to a nesting site? Two quantitative variables Summary Statistics: correlation Visualization: scatterplot Scatterplot A scatterplot is the graph of the relationship between two quantitative variables. Minitab: Graph -> Scatterplot -> Simple 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 Correlation The correlation is a measure of the strength and direction of linear association between two quantitative variables Sample correlation: r Population correlation: ρ ( rho ) r = for duration of dance and distance to site Minitab: Stat -> Basic Statistics -> Correlation r 1 Correlation 2. The sign indicates the direction of association 1. positive association: r > 0 2. negative association: r < 0 3. no linear association: r 0 3. The closer r is to ±1, the stronger the linear association 4. r has no units and does not depend on the units of measurement 5. The correlation between X and Y is the same as the correlation between Y and X 5

6 Correlation Guessing Game Enter PennState for the group ID. Highest scorer in the class by the Birst exam gets one extra credit point on Exam 1! z-score for Penalty Yards Correlation NFL Teams r = Malevolence Rating of Uniform Correlation Cautions 1. Correlation can be heavily affected by outliers. Always plot your data! Testosterone Levels and Time What is the correlation between testosterone levels and hour of the day? a) Positive b) Negative c) About 0 Are testosterone level and hour of the day associated? Correlation Cautions TVs and Life Expectancy 1. Correlation can be heavily affected by outliers. Always plot your data! 2. r = 0 means no linear association. The variables could still be otherwise associated. Always plot your data! Life Expectancy Mexico Sri Lanka China Morocco Egypt Vietnam Iraq Pakistan Yemen Cambodia Madagascar Haiti Uganda South Africa Angola Russia France Canada Australia Japan United Kingdom United States r = TVs per 1000 People 6

7 Correlation Cautions 1. Correlation can be heavily affected by outliers. Always plot your data! 2. r = 0 means no linear association. The variables could still be otherwise associated. Always plot your data! 3. Correlation does not imply causation! Summary: Two Quantitative Variables Summary Statistics: correlation Visualization: scatterplot To Do Read Sections 2.4 and 2.5 Do HW 2.2, 2.3, 2.4, 2.5 (due Friday, 9/18) 7