Statistics Lecture 3

zLast class: histograms, measures of center, percentiles and measures of spread zHave completed Sections zToday - Section 2.5 and begin Chapter 4 zSuggested Problems: 2.1, 2.48 (also compute mean), construct histogram of data in 2.48

Measures of Spread (cont.) z5 number summary often reported: yMin, Q 1, Q 2 (Median), Q 3, and Max ySummarizes both center and spread yWhat proportion of data lie between Q 1 and Q 3 ?

Box-Plot zDisplays 5-number summary graphically zBox drawn spanning quartiles zLine drawn in box for median zLines extend from box to max. and min values.

zCan compare distributions using side-by-side box-plots zWhat can you see from the plot?

Scatter-Plots zHelp assess whether there is a relationship between 2 continuous variables, zData are paired y(x 1, y 1 ), (x 2, y 2 ),... (x n, y n ) zPlot X versus Y zIf there is no natural pairing…probably not a good idea! zWhat sort of relationships might we see?

Sample Variance zSample variance of n observations: zCan be viewed as roughly the average squared deviation of observations from the sample mean zUnits are in squared units of data

Sample Standard Deviation zSample standard deviation of n observations: zCan be viewed as roughly the average deviation of observations from the sample mean zHas same units as data

zVariance and standard deviation are most useful when measure of center is zAs observations become more spread out, s : increases or decreases? zBoth measures sensitive to outliers

Empirical Rule for Bell-Shaped Distributions zApproximately y68% of the data lie in the interval y95% of the data lie in the interval zCan use these to help determine range of typical values or to identify potential outliers

Probability (Chapter 4) z“There is a 75% chance of rain tomorrow” zWhat does this mean?

Definitions zProbability of an outcome is a numerical measure of the chance of the outcome occurring zA experiment is random if its outcome is uncertain zSample space, S, is the collection of possible outcomes of an experiment zEvent is a set of outcomes zEvent occurs when one of its outcomes occurs

Example zA coin is tossed 2 times zS= zDescribe event of getting 1 heads and 1 tails

zProbability of an event is the long-term proportion of times the event would occur if the experiment is repeated many times zProbability of event, A is denoted P(A) z z P(A) is the sum of the probabilities for each outcomes in A zP(S)=1

zBag of balls has 5 red and 5 green balls z3 are drawn at random zS= zA is the event that at least 2 green are chosen zA= zP(A)=

Discrete Uniform Distribution zSample space has k possible outcomes S={e 1,e 2,…,e k } zEach outcome is equally likely zP(e i )= zIf A is a collection of distinct outcomes from S, P(A)=

Example (pg 140) zInherited characteristics are transmitted from one generation to the next by genes zGenes occur in pairs and offspring receive one from each parent zExperiment was conducted to verify this idea zPure red flower crossed with a pure white flower gives zTwo of these hybrids are crossed. Outcomes: zProbability of each outcome

zSometimes, not all outcomes are equally likely (e.g., fixed die) zRecall, probability of an event is long-term proportion of times the event occurs when the experiment is performed repeatedly zNOTE: Probability refers to experiments or processes, not individuals

Probability Rules zHave looked at computing probability for events zHow to compute probability for multiple events? zExample: 65% of Umich Business School Professors read the Wall Street Journal, 55% read the Ann Arbor News and 45% read both. A randomly selected Professor is asked what newspaper they read. What is the probability the Professor reads one of the 2 papers?

zAddition Rule: zIf two events are mutually exclusive: zComplement Rule

Independent Events zTwo events are independent if: zThe intuitive meaning is that the outcome of event A does not impact the probability of any outcome of event B