dates given in your syllabus.

Transcription

1 Slide 2-1

2 For exams (MD1, MD2, and Final): You may bring one 8.5 by 11 sheet of paper with formulas and notes written or typed on both sides to each exam. For the rest of the quizzes, you will take your quizzes in the recitation as before. So for the quiz dates we will go back to the original So for the quiz dates we will go back to the original dates given in your syllabus.

3 Chapter 2 Data

4 We treat variables as categorical or quantitative. Categorical variables identify a category for each case. Quantitative variables record measurements or amounts of something and must have units. Some variables can be treated as categorical or quantitative depending on what we want to learn from them.

5 What have we learned? (cont.) Case is an individual about whom or which we have the data. Population is all of the cases we wish we knew about. Sample is the cases we actually examine in seeking to understand the much larger population. Variable holds information about the same characteristic for many cases. Unit is a quantity or amount adopted as a standard measurement, such as dollars, or hours.

6 If the categories of your variables can be ordered, these variables are often called ordinal variables. Identifier variables are categorical variables with exactly one individual in each category. Examples: Social Security Number, ISBN, FedEx Tracking Number

7 Chapter 11 Understanding Randomness

8 What have we learned? An outcome is random if we know the possible values it can have, but not which particular value it takes. Randomizing makes is possible to generalize our findings to the population. A simulation model can help us investigate a question when we can t (or don t want to) collect data, and a mathematical answer is hard to calculate. Simulations can provide us with useful insights about the real world. Simulation component uses equally likely random digits to model simple random occurences whose outcomes may not be equally likely.

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10 Chapter 12 Sample Surveys

11 What have we learned? A representative sample can offer us important insights about populations. It s the size of the sample, not its fraction of the larger population, that determines the precision of the statistics it yields. There are several ways to draw samples, all based on the power of randomness to make them representative of the population of interest: Simple Random Sample, Stratified Sample, Cluster Sample, Systematic Sample, Multistage Sample Slide 1-11

12 What have we learned? (cont.) Bias can destroy our ability to gain insights from our sample: Nonresponse bias can arise when sampled individuals will not or cannot respond. Response bias arises when respondents answers might be affected by external influences, such as question wording or interviewer behavior. Slide 1-12

13 What have we learned? (cont.) Bias can also arise from poor sampling methods: Voluntary response samples are almost always biased and should be avoided and distrusted. Convenience samples are likely to be flawed for similar reasons. Even with a reasonable design, sample frames may not be representative. Undercoverage occurs when individuals from a subgroup of the population are selected less often than they should be. Slide 1-13

14 Chapter 3 Displaying and Describing Categorical Data

15 What have we learned? We can summarize categorical data by counting the number of cases in each category (expressing these as counts or percents). We can display the distribution in a bar chart or pie chart. And, we can examine two-way tables called contingency tables, examining marginal and/or conditional distributions of the variables. If conditional distributions of one variable are the same for every category of the other, the variables are independent. Slide 3-15

16 Pie Charts When you are interested in parts of the whole, a pie chart might be your display of choice. Pie charts show the whole group of cases as a circle. They slice the circle into pieces whose size is proportional to the fraction of the whole in each category. The angle of each slice= 360*rel. freq of corresponding category.

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18 Chapter 4 Displaying and Summarizing Quantitative Data

19 What have we learned? We can display the distribution of quantitative data with a histogram, stem-and-leaf display, or dotplot. We ve learned how to summarize distributions of quantitative variables numerically. Measures of center for a distribution include the mode, median and mean. Measures of spread include the range, IQR, and standard deviation. Use the median and IQR when the distribution is skewed. Use the mean and standard deviation if the distribution is symmetric.

20 Histograms: Earthquake Magnitudes (cont.) Histogram does not show individual values. But a stem-and-leaf plot does.

21 Stem-and-Leaf Example(cont.)

22 Stem-and-Leaf Example(cont.) The weights of 23 three-pound bags of apples are given as follows: use these data to construct a stem and leaf display for the weight data

23 Stem-and-Leaf Example(cont.) STEM LEAVES(unit=0.01) The weights of the bags range from 3.02 to 3.62, so can use as stems the values The leaves are determined by the digit found in the hundred s place of the original data.

24 Example-Dot plot The data below give the number of hurricane that happened each year from 1944 through 2000 as reported by science magazine. 3,2,1,2,4,3,7,2,3,3,2,5,2,2,4,2,2, 6,0,2,5,1,3,1,0,3,2,1,0,1,2,3,2,1, 2,2,2,3,1,1,1,3,0,1,3,2,1,2,1,1,0, 5,6,1,3,5,3

25 Dot plot for Hurricane data Dot plot for hurrican data C

26 Central tendency The central tendency of the set of measurements that is, the tendency of the data to cluster, or center, about certain numerical values. Central Tendency (Location) Center

27 Variability (Spread) The variability of the set of measurements that is, the spread of the data. Variation (Dispersion) Sample B Sample A Variation of Sample B Variation of Sample A

28 Mode Humps in a histogram are called modes. Mode is a value that occurs most often

29 Median (cont.) 1. Middle value in ordered sequence If n is odd, middle value of sequence If n is even, average of 2 middle values 2. Position of median in sequence Positioning Point = n + 1 2

30 Median Example Odd-Sized Sample Raw Data: Ordered: Position: Positioning Point Median = = n = =

31 Median Example Even-Sized Sample Raw Data: Ordered: Position: Positioning Point = Median = 2 n = = 2 2 =

32 Mean 1. Most common measure of central tendency 2. Acts as balance point 3. Affected by extreme values ( outliers ) x 4. Denoted where x n x i i= = = n x + x + + x n n Sample mean

33 Mean Example Raw Data: x n x i x + x + x + x + x + x i= = = n = =

34 Shape of a distribution

35 Shape Left-Skewed Mean Median Mode Symmetric Mean = Median= Mode Right-Skewed Mode Median Mean

36 Example Mean=45 Median=68 Mode=94 Is this data-set skewed? If it is, which direction is the Is this data-set skewed? If it is, which direction is the skewness?

37 Example Mean=45 Median=68 Mode=94 Is this data-set skewed? If it is, which direction is the Is this data-set skewed? If it is, which direction is the skewness? (Yes, skewed to the left)

38 Outlier An observation (or measurement) that is unusually large or small relative to the other values in a data set is called an outlier. Outliers affect statistical analyses. So you always check if data consist outlying observation/s.

39 Outlier Outliers typically are attributable to one of the following causes: 1. The measurement is observed, recorded, or entered into the computer incorrectly. 2. The measurement comes from a different population. 3. The measurement is correct but represents a rare (chance) event.

40 Sensitivity against the outliers Because the median considers only the order of values, it is resistant to values that are extraordinarily large or small; it simply notes that they are one of the big ones or small ones and ignores their distance from center. To choose between the mean and median, start by looking at the data. If the histogram is symmetric and there are no outliers, use the mean. However, if the histogram is skewed or with outliers, you are better off with the median.

41 Numerical Measures of Variability

42 They have same measure of central tendency, but their variations (spread) differ.

43 Spread: The Interquartile Range Range=Max-Min The interquartile range (IQR) lets us ignore extreme data values and concentrate on the middle of the data. IQR=Q3-Q1

44 Quartiles Split ordered data into 4 quarters 25% 25% 25% 25% Q 1 Q 2 Q 3 Lower quartile Q L (Q 1 ) is 25 th percentile. Middle quartile m (Q 2 ) is the median. Upper quartile Q U (Q 3 ) is 75 th percentile.

45 Quartile (Q 2 ) Example Raw Data: Ordered: Position: Q 2 is the median, the value with the position number 4= 8.9

46 Quartile (Q 1 ) Example Raw Data: Ordered: Position: Q L (Q 1 ) is median of bottom half = 6.3

47 Quartile (Q 3 ) Example Raw Data: Ordered: Position: Q U (Q 3 ) is median of bottom half = 11.7 IQR= =5.4

48 Spread: The Interquartile Range (cont.) The lower and upper quartiles are the 25 th and 75 th percentiles of the data, so The IQR contains the middle 50% of the values of the distribution, as shown in figure:

49 s = 2 i= 1 = Sample Variance Formula n ( x x ) i n ( x x ) + ( x x ) + L + ( x x ) 1 2 n 1 n n 1 in denominator!

50 A shortcut formula for variance

51 Sample Standard Deviation Formula s = = = s 2 n i= 1 ( x x ) i n ( x x ) + ( x x ) + L + ( x x ) 1 2 n 1 n

52 Standard Notation Measure Sample Population Mean x µ Standard Deviation s σ Variance s 2 σ 2 Size n N

53 Variance Example Raw Data: s s n n 2 ( ) i 2 i 1 i 1 2 = = = x x x = = where x = = 8. 3 n 1 n ( ) ( ) ( ) i

54 Thinking Challenge 1 Why do we need to take square root of variance to have a meaningful measure? Otherwise we would have a squared unit.

55 Thinking Challenge 2 Can the variance of a data set ever be negative? Can the variance ever be smaller than the standard deviation? No Yes

56 Thinking About Variation Since Statistics is about variation, spread is an important fundamental concept of Statistics. Measures of spread help us talk about what we don t know. When the data values are tightly clustered around the center of the distribution, the IQR and standard deviation will be small. When the data values are scattered far from the center, the IQR and standard deviation will be large.

57 Summary of Variation Measures Measure Formula Description Range X largest X smallest Total Spread Standard Deviation (Sample) Dispersion about Sample Mean Standard Deviation (Population) Variance (Sample) IQR n ( x i x) 2 i=1 n 1 n ( x i µ x ) 2 i=1 n N ( x i x) 2 i=1 n 1 Q 3 -Q 1 Dispersion about Population Mean Squared Dispersion about Sample Mean Spread for Middle Half

58 Box Plot The most extreme observation smaller than upper inner fence(q3+iqr*1.5=3.4)=2.64 Q3 Q2 Q1 The most extreme observation bigger than upper inner fence(q1-iqr*1.5=-0.2)=1.1

59 Box Plot 3. A second pair of fences, the outer fences, are defined at a distance of 3(IQR) from the hinges. One symbol (*) represents measurements falling between the inner and outer fences, and another (0) represents measurements beyond the outer fences. 4. Symbols that represent the median and extreme data points vary depending on software used. You may use your own symbols if you are constructing a box plot by hand.

60 Shape & Box Plot Left-Skewed Symmetric Right-Skewed Q 1 Median Q 3 Q 1 Median Q 3 Q Median Q 1 3

61 Rules for Detecting Quantitative Outliers Method Suspect Highly Suspect Box plot: z-score Values between inner and outer fences 2 < z < 3 Values beyond outer fences z > 3

62 Chapter 6 The Standard Deviation as a Ruler and the Normal Model

63 z Scores Describes the relative location of a measurement (x) compared to the rest of the data Sample z score Population z score z = x x s z = x µ σ Measures the number of standard deviations away from the mean a data value is located

64 What have we learned? With z-scores, we can compare values from different distributions or values based on different units, since z-scores have no units. z-scores can identify unusual or surprising values among data. z-scores measure the distance of each data value from the mean in standard deviations. A negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean.

65 What have we learned? (cont.) We ve learned that the Rule can be a useful rule of thumb for understanding distributions: For data that are unimodal and symmetric, about 68% fall within 1 SD of the mean, 95% fall within 2 SDs of the mean, and 99.7% fall within 3 SDs of the mean.

66 What have we learned? (cont.) Standardizing into z-scores does not change the shape of the distribution. Standardizing into z-scores changes the center by making the mean 0. Standardizing into z-scores changes the spread by making the standard deviation 1.

67 When Is a z-score BIG? A z-score gives us an indication of how unusual a value is because it tells us how far it is from the mean. Remember that a negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean. The larger a z-score is (negative or positive), the more unusual it is.

68 When Is a z-score Big? (cont.) Normal model (You may have heard of bellshaped curves. ). N(µ,σ) The N(0,1) model is called the standard Normal model (or the standard Normal distribution). Normal models are appropriate for distributions whose shapes are unimodal and roughly symmetric. These distributions provide a measure of how extreme a z-score is.

69 Ex Cars currently sold in the US have an average of 135 horsepower, with a standard deviation of 40 horsepower. What is the z- score for a car with 195 horse power? Z=( )/40=1.5

70 Ex People with z-scores greater than 2.5 on an IQ test are sometimes classified as geniuses. If IQ test scores have a mean of 100 and a std. dev. of 16 points, what IQ score do you need to be considered a genious? 2.5=(x-100)/16 x=140

71 The Rule (cont.) It turns out that in a Normal model: about 68% of the values fall within one standard deviation of the mean; about 95% of the values fall within two standard deviations of the mean; and, about 99.7% (almost all!) of the values fall within three standard deviations of the mean.

72 The Rule (cont.) The following shows what the Rule tells us:

73 Just Checking Suppose it takes you 20 minutes, on average, to drive to school, with a standard deviation of 2 minutes. Suppose a Normal model is appropriate for the distributions of driving times. A) How often will you drive at school less than 22 minutes? 84% of time B) How often will it take you more than 24 minutes? 2.5% of time

74 Finding Normal Percentiles by Hand (cont.) Table Z is the standard Normal table. We have to convert our data to z-scores before using the table. The figure shows us how to find the area to the left when we have a z-score of 1.80:

75 Finding Normal Percentiles Using Technology Using TI calculators Normalcdf(a,b,mean,std)

76 Finding Normal Percentiles Using Technology (cont.) normalcdf(-0.5,1,0,1)

77 From Percentiles to Scores: z in Reverse Example: What z-score represents the first quartile in a Normal model? P(Z<z 0 )=0.25 z 0 =? Or by TI calculator invnorm(p,mean,std) =invnorm(0.25,0,1)

78 Are You Normal? Normal Probability Plots (cont) A more specialized graphical display that can help you decide whether a Normal model is appropriate is the Normal probability plot. If the distribution of the data is roughly Normal, the Normal probability plot approximates a diagonal straight line. Deviations from a straight line indicate that the distribution is not Normal.

79 Are You Normal? Normal Probability Plots (cont) Nearly Normal data have a histogram and a Normal probability plot that look somewhat like this example: These two values are a bit lower than we d expect of the lowest two values in a Normal model.

80 Are You Normal? Normal Probability Plots (cont) A skewed distribution might have a histogram and Normal probability plot like this for which rule would not be accurate.

81 Ex In a standard Normal model, what value(s) of z cut(s) off the region described? A) The lowest 12% (invnorm(0.12,0,1)) B) The highest 30% 0.53 (invnorm(0.70,0,1)) C) The highest 7% 1.47 (invnorm(0.93,0,1)) D) The middle 50% (-0.67, 0.67) (invnorm(0.25,0,1))

82 Ex Based on the Normal model N(100,16) describing IQ scores, what percent of people s IQS would you expect to be A) Over 80? Z=(80-100)/16= = % or 0.5+normalcdf(80,100,100,16) B) Under 90? Z=(90-100)/16= The mean for the values of and -0.63=( )/2= % or 0.5-normalcdf(90,100,100,16) C) Between 112 and 132? or Normalcdf(112,132,100,16) Z1=( )/16=0.75 Z2=( )/16=2.00 The valu for 2.00-The value for 0.75= = %

83 Experiments & Sample Spaces In general, each occasion upon which we observe a random phenomenon is called a trial. At each trial, we note the value of the random phenomenon, and call it an outcome. The most basic outcome of a trial is a sample point. The collection of all possible outcomes is called the sample space.

84 Visualizing Sample Space 1. Listing for the experiment of tossing a coin once and noting up face S = {Head, Tail} Sample point 2. A pictorial method for presenting the sample space Venn Diagram H T S

85 Example Experiment: Tossing two coins and recording up faces: Is sample space as below? S={HH, HT, TT}

86 Tree Diagram 1 st coin H 2 nd coin T H T H T

87 Experiment Sample Space Examples Toss a Coin, Note Face Toss 2 Coins, Note Faces Sample Space {Head, Tail} {HH, HT, TH, TT} Select 1 Card, Note Kind {2, 2,..., A } (52) Select 1 Card, Note Color {Red, Black} Play a Football Game {Win, Lose, Tie} Inspect a Part, Note Quality {Defective, Good} Observe Gender {Male, Female}

88 Events 1. Specific collection of sample points 2. Simple Event Contains only one sample point 3. Compound Event Contains two or more sample points

89 What is Probability? 1. Numerical measure of the likelihood that event will occur P(Event) P(A) Prob(A) 1 Certain.5 2. Lies between 0 & 1 3. Sum of probabilities for all sample points in the sample space is 1 0 Impossible

90 Equally Likely Probability P(Event) = X / T X = Number of outcomes in the event T = Total number of sample points in Sample Space Each of T sample points is equally likely P(sample point) = 1/T T/Maker Co.

91 Thinking Challenge (sol.) Consider rolling two fair dice. Let event A=Having the sum of upfaces 6 or less. So, A={ (1,1), (1,2), (1,3), (1,4), (1,5), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2),(3,3), (4,1), (4,2), (5,1)} each with prob.1/36 P(A)=15/36=5/12

92 1. Union Unions & Intersections Outcomes in either events Aor B or both OR statement Denoted by symbol (i.e., A B) 2. Intersection Outcomes in both events Aand B AND statement Denoted by symbol (i.e., A B)

93 The table displays the probabilities for each of the six outcomes when rolling a particular unfair die. Suppose that the die is rolled once. Let A be the event that the number rolled is less than 4, and let B be the event that the number rolled is odd. Find P(A B). Outcome Probability A. 0.5 B. 0.2 C. 0.3 D. 0.7

94 Event Probability Using Two Way Table Event Event B 1 B 2 Total A 1 P(A 1 B 1 ) P(A 1 B 2 ) P(A 1 ) A 2 P(A 2 B 1 ) P(A 2 B 2 ) P(A 2 ) Total Joint Probability P(B 1 ) P(B 2 ) 1 Marginal (Simple) Probability

95 Complementary Events Complement of Event A The event that A does not occur All events not in A Denote complement of A by A C A A C S

96 3.4 The Additive Rule and Mutually Exclusive Events

97 Mutually Exclusive Events Example Experiment: Draw 1 Card. Note Kind & Suit. Sample Space: 2, 2, 2,..., A S Outcomes in Event Heart: 2, 3, 4,..., A Event Spade: 2, 3, 4,..., A Events and are Mutually Exclusive

98 Additive Rule 1. Used to get compound probabilities for union of events 2. P(A OR B) = P(A B) = P(A) + P(B) P(A B) 3. For mutually exclusive events: P(A OR B) = P(A B) = P(A) + P(B)

99 Thinking Challenge Let P(A)=0.25 and P(B C )=0.4. If P(A B)=0.85. Are the two events A, B mutually exclusive events? A. True B. False

100 Thinking Challenge Using the additive rule, what is the probability? 1. P(A D) = Event 2. P(B C) = Event C D Total A B Total

101 Solution* Using the additive rule, the probabilities are: 1. P(A D) = P(A) + P(D) P(A D) = + = P(B C) = P(B) + P(C) P(B C) = + =

102 Conditional Probability 1. Event probability given that another event occurred 2. Revise original sample space to account for new information Eliminates certain outcomes 3. P(A B) = P(A and B) = P(A B) P(B) P(B)

103 Thinking Challenge Using the table then the formula, what s the probability? 1. P(A D) = Event 2. P(C B) = Event C D Total A B Total

104 Solution* Using the formula, the probabilities are: P( A D)= P A B P D P( C B)= P C B P B 2 5 ( ) = = 2 ( ) P(D)=P(A D)+P(B D)=2/10+3/10 ( ) ( ) = = 1 4 P(B)=P(B D)+P(B C)=3/10+1/10

105 Multiplicative Rule 1. Used to get compound probabilities for intersection of events 2. P(A and B) = P(A B) = P(A) P(B A) = P(B) P(A B) 3. The key words both and and in the statement imply and intersection of two events, which in turn we should multiply probabilities to obtain the probability of interest.

106 Statistical Independence 1. Event occurrence does not affect probability of another event Toss 1 coin twice 2. Causality not implied 3. Tests for independence P(A B) = P(A) P(B A) = P(B) P(A B) = P(A) P(B)

107 Thinking Challenge 1) Consider a regular deck of 52 cards with two black suits i.e. (13), (13) and two red suits i.e. (13) (13). Given that you have a red card what is the probability that it is a queen? Also are the events getting a red card and getting a queen independent? A. 1/13, No (i.e. P(Queen) P(Queen Red)) B. 1/13, Yes (i.e. P(Queen)=P(Queen Red) C. 2/13, Yes (i.e. P(Queen) P(Queen Red) D. 2/13, No (i.e. P(Queen)=P(Queen Red) 2) Suppose that 23% of adults smoke cigarettes. Given a selected adult is a smoker, the probability that he/she has a lung condition before the age of 60 is 57%. What is the probability that a randomly selected person is a smoker and has a lung condition before the age of 60. A B C D. 0.13

108 Thinking Challenge 1) Consider a regular deck of 52 cards with two black suits i.e. (13), (13) and two red suits i.e. (13) (13). Given that you have a red card what is the probability that it is a queen? Also are the events getting a red card and getting a queen independent? A. 1/13, No (i.e. P(Queen) P(Queen Red)) B. 1/13, Yes (i.e. P(Queen)=P(Queen Red) C. 2/13, Yes (i.e. P(Queen) P(Queen Red) D. 2/13, No (i.e. P(Queen)=P(Queen Red) 2) Suppose that 23% of adults smoke cigarettes. Given a selected adult is a smoker, the probability that he/she has a lung condition before the age of 60 is 57%. What is the probability that a randomly selected person is a smoker and has a lung condition before the age of 60. A B C D. 0.13

109 Bayes s Rule P( B A)= P( A B)P( B) ( )P B C P( A B)P( B)+ P A B C ( ) Bayes s rule is useful for finding one conditional probability when other conditional probabilities are already known.

110 Bayes s Rule Example A company manufactures MP3 players at two factories. Factory I produces 60% of the MP3 players and Factory II produces 40%. Two percent of the MP3 players produced at Factory I are defective, while 1% of Factory II s are defective. An MP3 player is selected at random and found to be defective. What is the probability it came from Factory I?

111 Bayes s Rule Example 0.6 Factory I Defective Good 0.4 P(I D) = Factory II Defective Good P(I)P(D I) P(I)P(D I) + P(II)P(D II) = = 0.75

112 Random Variable A random variable is a variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point.

113 Random Variable (cont.) There are two types of random variables: Discrete random variables can take one of a finite number of distinct outcomes. Example: Number of credit hours Continuous random variables can take any numeric value within a range of values. Example: Cost of books this term

114 Random Variable A random variable is a variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point.

115 Random Variable (cont.) There are two types of random variables: Discrete random variables can take one of a finite number of distinct outcomes. Example: Number of credit hours Continuous random variables can take any numeric value within a range of values. Example: Cost of books this term

116 Discrete Probability Distribution The probability distribution of a discrete random variable is a graph, table, or formula that specifies the probability associated with each possible value the random variable can assume.

117 Requirements for the Probability Distribution of a Discrete Random Variable x 1. p(x) 0 for all values of x 2. Σ p(x) = 1 where the summation of p(x) is over all possible values of x.

118 a) It is not valid b) It is valid c) It is not valid d) It is not valid

119 a) {HHH,HTT,THT,TTH,THH,HTH,HHT,TTT} {0,1,2,3} b) {1/8,3/8,3/8,1/8} d) P(x=2 or x=3)= P(x=2)+P(x=3)=3/8+1/8=1/2

120 1. Expected Value (Mean of probability distribution) Weighted average of all possible values µ = E(x) = Σx p(x) 2. Variance Summary Measures Weighted average of squared deviation about mean σ 2 = E[(x µ) 2 ] = Σ (x µ) 2 p(x)=σx 2 p(x)-µ 2 3. Standard Deviation σ = 2 σ

121 Thinking challenge For the probability model given below, what is the value of P and E(X)? X P(x) P A. 0.1, 3.0 B. 0.2, 4.7 C. 0.3, 1.7 D. 0.4, 3.7

122 More About Means and Variances Adding or subtracting a constant from data shifts the mean but doesn t change the variance or standard deviation: E(X ± c) = E(X) ± c Var(X ± c) = Var(X) Example: Consider everyone in a company receiving a $5000 increase in salary.

123 More About Means and Variances (cont.) In general, multiplying each value of a random variable by a constant multiplies the mean by that constant and the variance by the square of the constant: E(aX) = ae(x) Var(aX) = a 2 Var(X) Stddev(aX)=aStddev(X) Example: Consider everyone in a company receiving a 10% increase in salary.

124 More About Means and Variances (cont.) In general, The mean of the sum of two random variables is the sum of the means. The mean of the difference of two random variables is the difference of the means. E(X ± Y) = E(X) ± E(Y) If the random variables are independent, the variance of their sum or difference is always the sum of the variances. Var(X ± Y) = Var(X) + Var(Y) Don t forget: Variances of independent random variables add. Standard deviations don t.