The empirical (68-95-99.7) rule With a bell shaped distribution, about 68% of the data fall within a distance of 1 standard deviation from the mean. 95% fall within 2 standard deviations of the mean. 99.7% fall within 3 standard deviations of the mean. What if the distribution is not bell-shaped? There is another rule, named Chebyshev's Rule, that tells us that there must be at least 75% of the data within 2 standard deviations of the mean, regardless of the shape, and at least 89% within 3 standard deviations. week3 1
Linear transformations A linear transformation changes the original value x into a new variable x new. x new is given by an equation of the form, xnew = a + bx Example 1.19 on page 54 in IPS. (i) A distance x measured in km. can be expressed in miles as follow,. xnew = 0.62x (ii) A temperature x measured in degrees Fahrenheit can be converted to degrees Celsius by x 5( 32) 160 5 new = x x 9 = 9 + 9 week3 2
Effect of a Linear Transformation Multiplying each observation in a data set by a number b multiplies both the measures of center (mean, median, and trimmed means) by b and the measures of spread (range, standard deviation and IQR) by b that is the absolute value of b. Adding the same number a to each observation in a data set adds a to measures of center, quartiles and percentiles but does not change the measures of spread. Linear transformations do NOT change the overall shape of a distribution. week3 3
Measure x x new Mean a + bx Median x M a+bm Mode Range IQR Stdev Mode R IQR s a+bmode b R b IQR b s week3 4
Example 1 A sample of 20 employees of a company was taken and their salaries were recorded. Suppose each employee receives a $300 raise in the salary for the next year. State whether the following statements are true or false. a) The IQR of the salaries will i. be unchanged ii. increase by $300 iii. be multiplied by $300 b) The mean of the salaries will i. be unchanged ii. increase by $300 iii. be multiplied by $300 week3 5
Nonlinear transformations A very common nonlinear transformation in statistic is the logarithm transformation. Recall: lnx = log e x where e is the natural number e = 2.7183. If measurements on a variable x have a right skewed distribution. The distribution of lnx will be roughly symmetric. If measurements on a variable x have a left skewed distribution. The distribution of lnx will be even more left skewed. week3 6
Example 2 - Nonlinear transformations Histogram for sales data Histogram for ln(sales) 200 60 50 Frequency 100 Frequency 40 30 20 10 0 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Sales 0 1 2 3 4 5 6 7 8 9 10 ln(sales) week3 7
Density curves Using software, clever algorithms can describe a distribution in a way that is not feasible by hand, by fitting a smooth curve to the data in addition to or instead of a histogram. The curves used are called density curves. It is easier to work with a smooth curve, because histogram depends on the choice of classes. Density Curve Density curve is a curve that is always on or above the horizontal axis. has area exactly 1 underneath it. A density curve describes the overall pattern of a distribution. week3 8
The area under the curve and above any range of values is the relative frequency (proportion) of all observations that fall in that range of values. Example: The curve below shows the density curve for scores in an exam and the area of the shaded region is the proportion of students who scores between 60 and 80. week3 9
Median and mean of Density Curve The median of a distribution described by a density curve is the point that divides the area under the curve in half. A mode of a distribution described by a density curve is a peak point of the curve, the location where the curve is highest. Quartiles of a distribution can be roughly located by dividing the area under the curve into quarters as accurately as possible by eye. week3 10
Normal distributions An important class of density curves are the symmetric unimodal bell-shaped curves known as normal curves. They describe normal distributions. All normal distributions have the same overall shape. The exact density curve for a particular normal distribution is specified by giving its mean μ and its standard deviation σ. The mean is located at the center of the symmetric curve and is the same as the median and the mode. Changing μ without changing σ moves the normal curve along the horizontal axis without changing its spread. week3 11
The standard deviation σ controls the spread of a normal curve. week3 12
There are other symmetric bell-shaped density curves that are not normal e.g. t distribution. The normal density curves are specified by a particular function. The height of a normal density curve at any point x is given by 2 1 x μ 1 e 2 σ σ 2π Notation: A normal distribution with mean μ and standard deviation σ is denoted by N(μ, σ). week3 13
The 68-95-99.7 rule In the normal distribution with mean μ and standard deviation σ, Approx. 68% of the observations fall within σ of the mean μ. Approx. 95% of the observations fall within 2σ of the mean μ. Approx. 99.7% of the observations fall within 3σ of the mean μ. week3 14
Example 1.23 on p72 in IPS The distribution of heights of women aged 18-24 is approximately N(64.5, 2.5), that is,normal with mean μ = 64.5 inches and standard deviation σ = 2.5 inches. The 68-95-99.7 rule says that the middle 95% (approx.) of women are between 64.5-5 to 64.5+5 inches tall. The other 5% have heights outside the range from 59.5 to 69.5 inches, and 2.5% of the women are taller than 69.5. Exercise: 1) The middle 68% (approx.) of women are between to inches tall. 2) % of the women are taller than 66.75. 3) % of the women are taller than 72. week3 15
Standardizing and z-scores If x is an observation from a distribution that has mean μ and standard deviation σ, the standardized value of x is given by z = A standardized value is often called a z-score. x σ μ A z-score tells us how many standard deviations the original observation falls away from the mean of the distribution. Standardizing is a linear transformation that transform the data into the standard scale of z-scores. Therefore, standardizing does not change the shape of a distribution, but changes the value of the mean and stdev. week3 16
Example 1.24 on p73 in IPS The heights of women is approximately normal with mean μ = 64.5 inches and standard deviation σ = 2.5 inches. The standardized height is The standardized value (z-score) of height 68 inches is or 1.4 std. dev. above the mean. A woman 60 inches tall has standardized height or 1.8 std. dev. below the mean. z = height 64.5 2.5 z = 68 64.5= 1.4 2.5 z = 60 64.5= 1.8 2.5 week3 17
The Standard Normal distribution The standard normal distribution is the normal distribution N(0, 1) that is, the mean μ = 0 and the sdev σ = 1. If a random variable X has normal distribution N(μ, σ), then the standardized variable Z = X μ σ has the standard normal distribution. Areas under a normal curve represent proportion of observations from that normal distribution. There is no formula to calculate areas under a normal curve. Calculations use either software or a table of areas. The table and most software calculate one kind of area: cumulative proportions. A cumulative proportion is the proportion of observations in a distribution that fall at or below a given value and is also the area under the curve to the left of a given value. week3 18
The standard normal tables Table A gives cumulative proportions for the standard normal distribution. The table entry for each value z is the area under the curve to the left of z, the notation used is P( Z z). e.g. P( Z 1.4 ) = 0.9192 week3 19
Standard Normal Distribution z.00.01.02.03.04.05.06.07.08.09 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0.5000.5040.5080.5120.5160.5199.5239.5279.5319.5359.5398.5438.5478.5517.5557.5596.5636.5675.5714.5753.5793.5832.5871.5910.5948.5987.6026.6064.6103.6141.6179.6217.6255.6293.6331.6368.6406.6443.6480.6517.6554.6591.6628.6664.6700.6736.6772.6808.6844.6879.6915.6950.6985.7019.7054.7088.7123.7157.7190.7224.7257.7291.7324.7357.7389.7422.7454.7486.7517.7549.7580.7611.7642.7673.7703.7734.7764.7794.7823.7852.7881.7910.7939.7967.7995.8023.8051.8078.8106.8133.8159.8186.8212.8238.8264.8289.8315.8340.8365.8389.8413.8438.8461.8485.8508.8531.8554.8577.8599.8621.8643.8665.8686.8708.8729.8749.8770.8790.8810.8830.8849.8869.8888.8907.8925.8944.8962.8980.8997.9015.9032.9049.9066.9082.9099.9115.9131.9147.9162.9177.9192.9207.9222.9236.9251.9265.9279.9292.9306.9319.9332.9345.9357.9370.9382.9394.9406.9418.9429.9441.9452.9463.9474.9484.9495.9505.9515.9525.9535.9545.9554.9564.9573.9582.9591.9599.9608.9616.9625.9633.9641.9649.9656.9664.9671.9678.9686.9693.9699.9706.9713.9719.9726.9732.9738.9744.9750.9756.9761.9767.9772.9778.9783.9788.9793.9798.9803.9808.9812.9817.9821.9826.9830.9834.9838.9842.9846.9850.9854.9857.9861.9864.9868.9871.9875.9878.9881.9884.9887.9890.9893.9896.9898.9901.9904.9906.9909.9911.9913.9916.9918.9920.9922.9925.9927.9929.9931.9932.9934.9936.9938.9940.9941.9943.9945.9946.9948.9949.9951.9952.9953.9955.9956.9957.9959.9960.9961.9962.9963.9964.9965.9966.9967.9968.9969.9970.9971.9972.9973.9974.9974.9975.9976.9977.9977.9978.9979.9979.9980.9981.9981.9982.9982.9983.9984.9984.9985.9985.9986.9986.9987.9987.9987.9988.9988.9989.9989.9989.9990.9990 The table shows area to left of z under standard normal curve For a negative number, -z : Area below (-z) = Area above (z) = 1 Area below (z) 20
The standard normal tables - Example What proportion of the observations of a N(0,1) distribution takes values a) less than z = 1.4? b) greater than z = 1.4? c) greater than z = -1.96? d) between z = 0.43 and z = 2.15? week3 21
Properties of Normal distribution If a random variable Z has a N(0,1) distribution then P(Z = z)=0. The area under the curve below any point is 0. The area between any two points a and b (a < b) under the standard normal curve is given by P(a Z b) = P(Z b) P(Z a) As mentioned earlier, if a random variable X has a N(μ, σ) distribution, then the standardized variable Z μ = X σ has a standard normal distribution and any calculations about X can be done using the following rules: week3 22
week3 23 P(X = k) = 0 for all k. The solution to the equation P(X k) = p is k = μ + σz p Where z p is the value z from the standard normal table that has area (and cumulative proportion) p below it, i.e. z p is the p th percentile of the standard normal distribution. ( ) = σ μ a Z P a X P ( ) = σ μ b Z P b X P 1 ( ) = σ μ σ μ b Z a P b X P a
Questions 1. The marks of STA221 students has N(65, 15) distribution. Find the proportion of students having marks (a) less then 50. (b) greater than 80. (c) between 50 and 80. 2. Example 1.30 on page 79 in IPS: Scores on SAT verbal test follow approximately the N(505, 110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT? 3. The time it takes to complete a stat220 term test is normally distributed with mean 100 minutes and standard deviation 14 minutes. How much time should be allowed if we wish to ensure that at least 9 out of 10 students (on average) can complete it? (final exam Dec. 2001) week3 24
4. General Motors of Canada has a deal: an oil filter and lube job in 25 minutes or the next one free. Suppose that you worked for GM and knew that the time needed to provide these services was approximately normal with mean 15 minutes and std. dev. 2.5 minutes. How many minutes would you have recommended to put in the ad above if it was decided that about 5 free services for 100 customers was reasonable? 5. In a survey of patients of a rehabilitation hospital the mean length of stay in the hospital was 12 weeks with a std. dev. of 1 week. The distribution was approximately normal. a) Out of 100 patients how many would you expect to stay longer than 13 weeks? b) What is the percentile rank of a stay of 11.3 weeks? c) What percentage of patients would you expect to be in longer than 12 weeks? d) What is the length of stay at the 90 th percentile? e) What is the median length of stay? week3 25
Normal quantile plots and their use A histogram or stem plot can reveal distinctly nonnormal features of a distribution. If the stem-plot or histogram appears roughly symmetric and unimodal, we use another graph, the normal quantile plot as a better way of judging the adequacy of a normal model. Any normal distribution produces a straight line on the plot. Use of normal quantile plots: If the points on a normal quantile plot lie close to a straight line, the plot indicates that the data are normal. Systematic deviations from a straight line indicate a nonnormal distribution. Outliers appear as points that are far away from the overall pattern of the plot. week3 26
Histogram, the nscores plot and the normal quantile plot for data generated from a normal distribution (N(500, 20)). 15 540 530 10 520 Frequency 5 value 510 500 490 480 0 460 470 480 490 500 510 520 530 540 value 470 460 Normal Probability Plot for value -2-1 0 1 2 ncores 99 ML Estimates 95 90 Mean: StDev: 500.343 17.4618 Percent 80 70 60 50 40 30 20 10 5 1 week3 27 450 500 550 Data
Histogram, the nscores plots and the normal quantile plot for data generated from a right skewed distribution 10 Frequency 5 0 0 5 10 value 10 value 5 0-2 -1 0 1 2 week3 ncores 21 28
2 1 ncores 0-1 -2 0 5 10 value Norm al Probability Plot for value 99 M L Estim ates 95 90 M ean: StDev: 2.64938 2.17848 Percent 80 70 60 50 40 30 20 10 5 1 0 5 10 week3 29 Data
Histogram, the nscores plots and the normal quantile plot for data generated from a left skewed distribution 10 Frequency 5 0 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 value 1.0 0.9 0.8 value 0.7 0.6 0.5 0.4 0.3-2 -1 0 1 2 nscore week3 30
2 1 nscore 0-1 -2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 value Normal Probability Plot for value 99 M L Estimates 95 90 M ean: StDev: 0.8102 0.161648 Percent 80 70 60 50 40 30 20 10 5 1 0.50 0.75 1.00 1.25 Data week3 31
Histogram, the nscores plots and the normal quantile plot for data generated from a uniform distribution (0,5) Frequency 9 8 7 6 5 4 3 2 1 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 value 5 4 value 3 2 1 0-2 -1 0 1 2 ncores week3 32
2 1 ncores 0-1 -2 0 1 2 3 4 5 value Normal Probability Plot for value 99 M L Estim ates 95 90 M ean: StDev: 2.21603 1.46678 Percent 80 70 60 50 40 30 20 10 5 1-2 -1 0 1 2 3 4 5 6 week3 33 Data
Question (similar to Q5 Term test Oct, 2000) Below are 4 normal probability (quantile) plots and 4 histograms produced by MINITAB for some data sets. The histograms are not in the same order as normal scores plots. Match the histograms with the nscores plots. week3 34
120 10 110 data 100 90 Frequency 5 80 0-2 -1 0 1 2 nscores 0 2 4 6 8 10 12 14 data 40 50 40 data 30 Frequency 30 20 10 20 0-2 -1 0 1 2 nscores 0 10 20 30 40 50 60 data 14 12 15 10 data 8 6 4 Frequency 10 5 2 0-2 -1 0 1 2 nscores 0 80 84 88 92 96 100 104 108 112 116 data 60 8 data 50 40 30 20 10 0-2 -1 0 1 2 nscores Frequency 7 6 5 4 3 2 1 0 week3 20 22 24 26 28 30 32 34 36 38 40 35 data
Looking at data - relationships Two variables measured on the same individuals are associated if some values of one variable tend to occur more often with some values of the second variable than with other values of that variable. When examining the relationship between two or more variables, we should first think about the following questions: What individuals do the data describe? What variables are present? How are they measured? Which variables are quantitative and which are categorical? Is the purpose of the study is simply to explore the nature of the relationship, or do we hope to show that one variable can explain variation in the other? week3 36
Response and explanatory variables A response variable measure an outcome of a study. An explanatory variable explains or causes changes in the response variables. Explanatory variables are often called independent variables and response variables are called dependent variables. The ides behind this is that response variables depend on explanatory variables. We usually call the explanatory variable x and the response variable y. week3 37
Scatterplot A scatterplot shows the relationship between two quantitative variables measured on the same individuals. Each individual in the data appears as a point in the plot fixed by the values of both variables for that individual. Always plot the explanatory variable, if there is one, on the horizontal axis (the x axis) of a scatterplot. Examining and interpreting Scatterplots Look for overall pattern and striking deviations from that pattern. The overall pattern of a scatterplot can be described by the form, direction and strength of the relationship. An important kind of deviation is an outlier, an individual value that falls outside the overall pattern. week3 38
Example There is some evidence that drinking moderate amounts of wine helps prevent heart attack. A data set contain information on yearly wine consumption (litters per person) and yearly deaths from heart disease (deaths per 100,000 people) in 19 developed nations. Answer the following questions. What is the explanatory variable? What is the response variable? Examine the scatterplot below. week3 39
300 200 100 0 1 2 3 4 5 6 7 8 9 Wine week3 40 Heart disease deaths
Interpretation of the scatterplot The pattern is fairly linear with a negative slope. No outliers. The direction of the association is negative. This means that higher levels of wine consumption are associated with lower death rates. This does not mean there is a causal effect. There could be lurking variables. For example, higher wine consumption could be linked to higher income, which would allow better medical care. MINITAB command for scatterplot Graph > Plot week3 41
Categorical variables in scatterplots To add a categorical variable to a scatterplot, use a different colour or symbol for each category. The scatterplot below shows the relationship between the world record times for 10,000m run and the year for both men and women. 2300 2200 F M Time (seconds) 2100 2000 1900 1800 1700 1600 1900 1950 Year 2000 week3 42
Categorical explanatory variables Scatterplots display the association between two quantitative variables. To display a relationship between a categorical explanatory variable and a quantitative response variable, make a side-byside comparison of the distributions of the response for each category. A back-to-back stemplot compares two distributions. Side-by-side boxplots compare any number of distributions. week3 43
Example We want to investigate to association between how much education a person has and his/her income. Education appears as a categorical variable. 1 = did not reach high school, 2 = some high school but no high school diploma. up to 6 = postgraduate degree. Order the categories and make side-by side boxplots for the income. week3 44
The side-by-side boxplots show a strong positive association between education and earnings. week3 45
week3 46 Correlation A sctterplot displays the form, direction and strength of the relationship between two quantitative variables. Correlation (denoted by r) measures the direction and strength of the liner relationship between two quantitative variables. Suppose that we have data on variables x and y for n individuals. The correlation r between x and y is given by ( ) y x n i i i n i y i x i s s nxy y x n s y y s x x n r = = = = 1 1 1 1 1 1
Example Family income and annual savings in thousand of $ for a sample of eight families are given below. savings income C3 C4 C5 1 36-1.42887-1.45101 2.07331 2 39-1.02062-1.03144 1.05271 2 42-0.61237-0.61187 0.37469 5 45-0.20412-0.19230 0.03925 5 48 0.20412 0.22727 0.04433 6 51 0.61237 0.64684 0.39611 7 54 1.02062 1.06641 1.08840 8 56 1.42887 1.34612 1.92343 Sum of C5 = 6.99429 r = 6.99429/7 = 0.999185 MINITAB command: Stat > Basic Statistics > Correlation week3 47
Properties of correlation Correlation requires both variables to be quantitative and make no use of the distinction between explanatory and response variables. Because r uses standardized values of observations, it does not depend on units of measurements of x and y. Correlation r has no unit if measurement. Positive r indicates positive association between the variables and negative r indicates negative association. Correlation measures the strength of only the linear relationship between two variables, it does not describe curved relationship! r is always a number between 1 and 1. Values of r near 0 indicates a weak linear relationship. The strength of the linear relationship increases as r moves away from 0. Values of r close to 1 or 1 indicates that the points lie close to a straight line. r is not resistant. r is strongly affected by a few outliers. week3 48
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Question from Term test, summer 99 MINITAB analyses of math and verbal SAT scores is given below. Variable N Mean Median TrMean StDev SE Mean Verbal 200 595.65 586.00 595.57 73.21 5.18 Math 200 649.53 649.00 650.37 66.35 4.69 GPA 200 2.6300 2.6000 2.6439 0.5803 0.0410 Variable Minimum Maximum Verbal 361.00 780.00 Math 441.00 800.00 GPA 0.3000 3.9000 Stem-and-leaf of Verbal N = 200 Leaf Unit = 10 1 3 6 4 4 034 19 4 566888888889999 52 5 000000122222222333333333444444444 (56) 5 55555555555556666666777777777777778888888888888889999999 92 6 00000000011111111222222333333333444444444444444 45 6 555555666666666778888888889999 15 7 0011112244 5 7 55568 week3 50
Stem-and-leaf of Math N = 200 Leaf Unit = 10 1 4 4 3 4 79 12 5 001222234 38 5 55555666677777778888889999 (63) 6 000000000000001111111111112222222222222222333333333344444444444 99 6 555555555666666666666667777777777788888889999999 51 7 000000000011111111111112222222333334444 12 7 5566777789 2 8 00 30 20 20 Frequency 10 Frequency 10 0 0 400 500 600 700 800 Math 400 500 600 700 800 Verbal week3 51
a) Find the 25 th percentile, 75 th percentile and the IQR of the math SAT scores. b) You were one of the students of this study and your math SAT score was 532. What is your z-score and percentile standing? c) If the math SAT scores were in fact left (negatively) skewed, but the mean was still 650, what could you say about the percentile standing of someone who obtains a score of 650? d) What is the class width? i) of the histogram for verbal SAT scores? ii) of the stemplot of the verbal SAT scores? e) Describe both the verbal and math score distributions and compare one with the other. week3 52
g) Give a rough sketch of how a normal probability plot would look if the verbal scores were i. Right (positively) skewed ii. Uniform in shape h) For verbal scores, aside from running through the data and tallying, can you determine the approx. percentage of scores which fall between 523 and 668? If so give the percentage. week3 53
Question (Term Test May 98) Descriptive statistics of scores of 3 groups of students are given below. Variable Group N Mean Median TrMean StDev Post1 B 22 6.682 6.500 6.650 2.767 D 22 9.773 10.000 9.800 2.724 S 22 7.773 7.000 7.750 3.927 Using the information above estimate the following in some reasonable way. State any assumptions that you have to make. (a) The 90 th percentile of the post1 scores using method B. b) The proportion of post1 scores that would be 7 or higher for those using method D. week3 54