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MATH 3339 Statistics for the Sciences sec 2.3; 2.6 Wendy Wang, Ph.D. wwang60@central.uh.edu Lecture 4 - 3339 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 1 / 44
Outline 1 Understanding Standard Deviation 2 Calculating The Standard Deviation 3 Measures of Variability 4 Jointly Distributed Data 5 Scatterplots for Jointly Distributed Variables 6 Covariance and Correlation Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 2 / 44
Measuring Spread: The Standard Deviation Measures spread by looking at how far the observations are from their mean. Most common numerical description for the spread of a distribution. A larger standard deviation implies that the values have a wider spread from the mean. Denoted s when used with a sample. This is the one we calculate from a list of values. Denoted σ when used with a population. This is the "idealized" standard deviation. The standard deviation has the same units of measurements as the original observations. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 3 / 44
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Definition of the Standard Deviation The standard deviation is the average distance each observation is from the mean. Using this list of values from a sample: 3, 3, 9, 15, 15 The mean is __. By definition, the average distance each of these values are from the mean is __. So the standard deviation is __. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 4 / 44
Values of the Standard Deviation The standard deviation is a value that is greater than or equal to zero. It is equal to zero only when all of the observations have the same value. By the definition of standard deviation determine s for the following list of values. ▶ 2, 2, 2, 2 : standard deviation = __ ▶ 125, 125, 125, 125, 125: standard deviation = __ Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 5 / 44
Adding or Subtracting a Value to the Observations Adding or subtracting the same value to all the original observations does not change the standard deviation of the list. Using this list of values: 3, 3, 3, 15, 15, 15 mean = 9, standard deviation = __. If we add 4 to all the values: 7, 7, 7, 19, 19, 19 mean = 13 , standard deviation = __ Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 6 / 44
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Multiplying or Dividing a Value to the Observations Multiplying or dividing the same value to all the original observations will change the standard deviation by that factor. Using this list of values: 3, 3, 3, 15, 15, 15: mean = 9, standard deviation = 6. If we double all the values: 6, 6, 6, 30, 30, 30 mean = 18 , standard deviation = __ Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 7 / 44
Population Variance and Standard Deviation If N is the number of values in a population with mean mu , and x i represents each individual in the population, then the population variance is found by: σ 2 = ∑ N i = 1 ( x i − µ ) 2 N and the population standard deviation is the square root, σ = √ σ 2 . Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 8 / 44
Sample Variance and Standard Deviation Most of the time we are working with a sample instead of a population. So the sample variance is found by: s 2 = ∑ n i = 1 ( x i − ¯ x ) 2 n − 1 and the sample standard deviation is the square root, s = √ s 2 . Where n is the number of observations (samples), x i is the value for the i th observation and ¯ x is the sample mean. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 9 / 44
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Calculating the Standard Deviation By Hand When calculating by hand we will calculate s . 1. Find the mean of the observations ¯ x . 2. Calculate the difference between the observations and the mean for each observation x i − ¯ x . This is called the deviations of the observations. 3. Square the deviations for each observation ( x i − ¯ x ) 2 . 4. Add up the squared deviations together ∑ n i = 1 ( x i − ¯ x ) 2 . 5. Divide the sum of the squared deviations by one less than the number of observations n − 1. This is the variance s 2 = 1 n − 1 n X i = 1 ( x i − ¯ x ) 2 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 10 / 44
Step 6: Standard Deviation 6. Find the square root of the variance. This is the standard deviation s = v u u t 1 n − 1 n X i = 1 ( x i − ¯ x ) 2 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 11 / 44
Example: Section A Determine the sample standard deviation of the test scores for Section A. Section A Scores ( X i ) 65 66 67 68 71 73 74 77 77 77 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 12 / 44
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Step 1: Calculate the Mean The sample mean is ¯ x = 71 . 5. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 13 / 44
Use Table To Calculate Standard Deviation Variable Deviations Deviations Squared Score ( X i ) X i − ¯ X ( X i − ¯ X ) 2 65 66 67 68 71 73 74 77 77 77 sum Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 14 / 44
Step 2: Calculate Deviations For All Values Variable Deviations Deviations Squared Score ( X i ) X i − ¯ X ( X i − ¯ X ) 2 65 65 − 71 . 5 = − 6 . 5 66 66 − 71 . 5 = − 5 . 5 67 67 − 71 . 5 = − 4 . 5 68 68 − 71 . 5 = − 3 . 5 71 71 − 71 . 5 = − 0 . 5 73 73 − 71 . 5 = 1 . 5 74 74 − 71 . 5 = 2 . 5 77 77 − 71 . 5 = 5 . 5 77 77 − 71 . 5 = 5 . 5 77 77 − 71 . 5 = 5 . 5 sum Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 15 / 44
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Step 3: Calculate Squared Deviations Variable Deviations Deviations Squared Score ( X i ) X i − ¯ X ( X i − ¯ X ) 2 65 65 − 71 . 5 = − 6 . 5 ( − 6 . 5 ) 2 = 42 . 25 66 66 − 71 . 5 = − 5 . 5 ( − 5 . 5 ) 2 = 30 . 25 67 67 − 71 . 5 = − 4 . 5 ( − 4 . 5 ) 2 = 20 . 25 68 68 − 71 . 5 = − 3 . 5 ( − 3 . 5 ) 2 = 12 . 25 71 71 − 71 . 5 = − 0 . 5 ( − 0 . 5 ) 2 = 0 . 25 73 73 − 71 . 5 = 1 . 5 1 . 5 2 = 2 . 25 74 74 − 71 . 5 = 2 . 5 2 . 5 2 = 6 . 25 77 77 − 71 . 5 = 5 . 5 5 . 5 2 = 30 . 25 77 77 − 71 . 5 = 5 . 5 5 . 5 2 = 30 . 25 77 77 − 71 . 5 = 5 . 5 5 . 5 2 = 30 . 25 sum Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 16 / 44
Step 4: Calculate the Sum of the Squared Deviations Variable Deviations Deviations Squared Score( X i ) X i − ¯ X ( X i − ¯ X ) 2 65 65 − 71 . 5 = − 6 . 5 ( − 6 . 5 ) 2 = 42 . 25 66 66 − 71 . 5 = − 5 . 5 ( − 5 . 5 ) 2 = 30 . 25 67 67 − 71 . 5 = − 4 . 5 ( − 4 . 5 ) 2 = 20 . 25 68 68 − 71 . 5 = − 3 . 5 ( − 3 . 5 ) 2 = 12 . 25 71 71 − 71 . 5 = − 0 . 5 ( − 0 . 5 ) 2 = 0 . 25 73 73 − 71 . 5 = 1 . 5 1 . 5 2 = 2 . 25 74 74 − 71 . 5 = 2 . 5 2 . 5 2 = 6 . 25 77 77 − 71 . 5 = 5 . 5 5 . 5 2 = 30 . 25 77 77 − 71 . 5 = 5 . 5 5 . 5 2 = 30 . 25 77 77 − 71 . 5 = 5 . 5 5 . 5 2 = 30 . 25 sum ∑ n i = 1 ( X i − ¯ X ) 2 = 204.5 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 17 / 44
Step 5: Calculate the Variance variance = s 2 = 1 n − 1 n X i = 1 ( x i − ¯ x ) 2 = 204 . 5 9 = 22 . 7222 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 18 / 44
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Step 6: Take the Square Root of the Variance standard deviation = s = v u u t 1 n − 1 n X i = 1 ( x i − ¯ x ) 2 = √ 22 . 7222 = 4 . 77 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 19 / 44
Sample Standard Deviation of Section A test scores Sample standard deviation is s = 4.77. This implies that from the sample of the 10 students from section A the tests scores has a spread, on average, of 4.77 points from the mean of 71.50 points. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 20 / 44
Multiply by 2, You try, Question 1 For the following dataset the mean is ¯ x = 4 . 5, the variance is s 2 = 3 . 5 and the standard deviation is s = 1 . 870829. 3, 6, 2, 7, 4, 5 Now, multiply each value by 2. What is the new mean, new variance and the new standard deviation? a) 4.5, 14, 3.5 b) 4.5, 3.5, 1.870829 c) 9, 14, 3.7416 d) 9, 7, 3.7416 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 21 / 44
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Calculating Standard Deviation For larger data sets use a calculator or computer software. Each calculator is different if you cannot determine how to compute standard deviation from your calculator ask your instructor. For this course we will be using R as the software. The function for the sample standard deviation in R is sd( data name$variable name ) . Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 22 / 44
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You try, Question 2 2. This is a standard deviation contest, which list of numbers have the largest standard deviation? No calculations are required. a) 10, 10, 10, 10 b) 20, 20, 20, 20 c) 10, 10 ,20, 20 d) 10, 15, 15, 20 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 23 / 44
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Measures of Variability: Coefficient of Variation This is to compare the variation between two groups. The coefficient of variation (cv) is the ratio of the standard deviation to the mean. cv = sd mean A smaller ratio will indicate less variation in the data. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 24 / 44
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CV of test scores Section A Section B Sample Size 10 10 Sample Mean 71.5 71.5 Sample Standard Deviation 4.770 18.22 CV 4 . 77 71 . 5 = 0 . 066 18 . 22 71 . 5 = 0 . 2548 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 25 / 44
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CV Example The following statistics were collected on two different groups of stock prices: Portfolio A Portfolio B Sample size 10 15 Sample mean $52.65 $49.80 Sample standard deviation $6.50 $2.95 What can be said about the variability of each portfolio? Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 26 / 44
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Jointly Distributed Data Motivating examples In the housing market, for a larger size house, the price of the house increases. In a stadium, is the number of hot dogs sold related to the number of sodas sold? A survey wants to know if there is a relationship between age and health care cost. An insurance wants know if there is a relationship between color of a car and the number of accidents. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 27 / 44
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Jointly Distributed Data Jointly Distributed Data is data for two different variables and we want to know the relationship between these two variables. Several applications look at the relationship between two variables. The data can be quantitative or categorical. If both variables are categorical we can look at bar graphs and cross tabulations to determine if there is a relationship. If one variable is quantitative and another variable is categorical we can use the side-by-side box plot to look at the relationship. If both variables are quantitative we look at the scatter plot to determine if there is a relationship. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 28 / 44
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Two Jointly Distributed Quantitative Variables A response variable measures an outcome of a study. Sometimes called a dependent variable. Usually the y -variable. An explanatory variable explains or influences changes in a response variable. Sometimes called an independent variable or predictor or factor. Usually the x -variable Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 29 / 44
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Types of variables In the housing market for a larger size house, the price of the house increases. The response variable is the price of the house. The explanatory variable is the size of the house. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 30 / 44
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You try For each of the scenarios determine the response variable. 3. A survey wants to know if there is a relationship between age and health care cost. a) age b) health care cost c) Not enough information 4. In a stadium, is the number of hot dogs sold related to the number of sodas sold? a) Number of hot dogs sold b) Number of sodas sold c) Not enough information Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 31 / 44
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Plotting the data: Scatterplots The best way to initially observe the relationship between two quantitative variables measured on the same individuals. The values of one variable appear on the horizontal axis, and the values of the other variable appear on the vertical axis. Each individual in the data appears as the point in the plot fixed by the values of both variables for that individual. Scatterplots can show if there is some kind of association between the two quantitative variables. To create a scatterplot in R plot(explanatory,response) Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 32 / 44
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Example We want to look at the relationship between the estimated cost of the car and age of the vehicle. Vehicle Estimated Cost Age Honda Insight $5,555 8 Toyota Prius $17,888 3 Toyota Prius $9,963 6 Toyota Echo $6,793 5 Honda Civic Hybrid $10,774 5 Honda Civic Hybrid $16,310 2 Chevrolet Prizm $2,475 8 Mazda Protege $2,808 10 Toyota Corolla $7,073 9 Acura Integra $8,978 8 Scion xB $11,213 2 Scion xA $9,463 3 Mazda3 $15,055 2 Mini Cooper $20,705 2 Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 33 / 44
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Scatterplot 2 4 6 8 10 5000 10000 15000 20000 Age Cost Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 34 / 44
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Creating a Scatterplot in R plot(x,y) > plot(carsreg$Age,carsreg$Cost) Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 35 / 44
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Interpreting scatterplots Look for Direction : A pattern that runs from the upper left to the lower right is said to have a negative direction. A trend that is running the other way has a positive direction. Look for Form : If there is a straight line relationship, it will appear as a cloud or swarm of points stretched out in a generally consistent, straight form. This is linear form. Look for Strength : How much scatter the plot has. Do the points appear to follow a single stream? This is a very strong association. Or does the swarm of points seem to form a vague cloud through which we can barely discern any trend or pattern? This is a weak association. Look for the unexpected. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 36 / 44
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Example 2 Suppose we want to know if there is an association between the number of spaces a property is from GO and the cost of the property in a monopoly game. 0 10 20 30 40 50 100 150 200 250 300 350 400 Spaces Cost Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 37 / 44
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Example 3 Suppose we want to know if there is an association between the height of the father and their child. 65 70 75 60 65 70 75 father's height child's height Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 38 / 44
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Covariance and Correlation If we have two quantitative variables from a sample, then their covariance is calculated by: cov ( x , y ) = 1 n − 1 n X i = 1 ( x i − ¯ x )( y i − ¯ y ) In R: cov(x,y) > cov(carsreg$Age,carsreg$Cost) [1] -13254.98 However, the strength of the relationship cannot be measured by the covariance, thus we divide each deviation in the sum by the standard deviation of that variable. The result is call the correlation cor ( x , y ) = cov ( x , y ) sd ( x ) ∗ sd ( y ) In R: cor(x,y) Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 39 / 44
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Facts About Correlation: r The correlation coefficient is a value that measures the direction and strength of a straight-line relationship between two quantitative variables. Correlation makes no distinction between explanatory and response variables. Correlation requires that both variables be quantitative. Because r uses the standardize values of the observations, r does not change when we change the units of measurement of x , y or both. Positive r indicates positive association between the variables, and negative r indicates negative association. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 40 / 44
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Facts About Correlation: r The r is always a number between − 1 and 1. ▶ Values close to 0 indicate a very weak linear relationship. ▶ If r is close to − 1 the association is a very strong negative linear relationship. ▶ If r is close to 1 the association is a very strong positive association. Correlation measures the strength of only a linear relationship. r is strongly affected by a few outlying observations. Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 41 / 44
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r = -0.078 160 140 120 100 80 60 40 20 0 85 80 75 70 65 60 55 50 Income Height Scatterplot of Height vs Income Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 42 / 44
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r = -0.707 42000 40000 38000 36000 34000 32000 5 4 3 2 1 Price Reliability Scatterplot of Reliability vs Price Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 43 / 44
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r = 0.925 140 120 100 80 60 40 20 500 400 300 200 100 0 Speed Height Scatterplot of Height vs Speed Wendy Wang, Ph.D. wwang60@central.uh.edu MATH 3339 Lecture 4 - 3339 44 / 44
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