The Normal Distribution. John McGready, PhD Johns Hopkins University

Transcription

1 The Normal Distribution John McGready, PhD Johns Hopkins University

2 General Properties of The Normal Distribution The material in this video is subject to the copyright of the owners of the material and is being provided for educational purposes under rules of fair use for registered students in this course only. No additional copies of the copyrighted work may be made or distributed.

3 Learning Objectives Upon completion of this lecture, you will be able to: Describe the basic properties of the normal curve Describe how the normal distribution is completely defined by its mean and standard deviation Recite the % rule for the normal distribution with regards to standard deviations 3

4 The Normal Distribution 1 The normal distribution is a theoretical probability distribution that is perfectly symmetric about its mean (and median and mode) A bell -like shape 4

5 The Normal Distribution 2 The normal distribution is also called the Gaussian distribution in honor of its inventor Carl Friedrich Gauss 5

6 Defining Quantities for any Normal Distribution Normal distributions are uniquely defined by two quantities: a mean (µ) and standard deviation (σ) There are literally an infinite number of possible normal curves for every possible combination of (µ) and (σ) 6

7 Underlying Formula for Normal Distribution This function defines the normal curve for any given (µ) and (σ) The proportion of values falling between a and b under a normal curve is given by: aa bb 1 2ππσσ ee (xx μμ)2 2σσ 2 dddd 7

8 Structural Properties of the Normal Distribution (Curve) 1 All normal distributions, regardless of mean and standard deviation values, have the same structural properties: Mean = median (= mode) Values are symmetrically distributed around the mean Values closer to the mean are more frequent than values farther from the mean 8

9 Structural Properties of the Normal Distribution (Curve) 2 All normal distributions, regardless of mean and standard deviation values, have the same structural properties: The entire distribution of values described by a normal distribution can be completely specified by knowing just the mean and standard deviation Since all normal distributions have the same structural properties, we can use a reference distribution, called the standard normal distribution, to elaborate on some of these properties In the next section, we ll show that any normal distribution can be easily rescaled to this standard normal distribution 9

10 The Rule for the Normal Distribution 1 68% of the observations in a normal distribution fall within one standard deviation of the mean 10

11 The Rule for the Normal Distribution 2 There are several ways to state this. For data whose distribution is approximately normal: 68% of the observations fall within one standard deviation of the mean The probability that any randomly selected value is within one standard deviation of the mean is 0.68 or 68% 11

12 The Rule for the Normal Distribution 3 95% of the observations fall within two standard deviations of the mean (truthfully, within 1.96) 12

13 The Rule for the Normal Distribution % of the observations fall within three standard deviations of the mean 13

14 2.5 th and 97.5 th Percentiles of a Normal Distribution 95% of the observations fall within two standard deviations of the mean (truthfully, within 1.96) The middle 95% of values fall between μμ -2σ and μμ+2σ 2.5% of the values are smaller than (and hence 97.5% are greater than) μμ -2σ 97.5% of the values are smaller than (and hence 2.5% are greater than) μμ+2σ 14

15 Percentage of Observations Under the Normal Distribution Where did this rule come from: in other words, how do I know these relationships? What about the percentages under the curve for other standard deviation distances from the mean? All of the information I quoted, and much more, can be found in a standard normal table 15

16 The Standard Normal Distribution The standard normal distribution is a normal distribution with mean μμ =0, and standard deviation σ=1 Any normal distribution with mean μμ and standard deviation σ can be rescaled to a standard normal distribution. 16

17 Percentage of Observations Under the Normal Distribution: Exhibit A Source: 17

18 Percentage of Observations Under the Normal Distribution: Exhibit B Source: 18

19 Percentage of Observations Under the Normal Distribution 2 In this class, I will only have you find relevant percentages under a normal curve for some early activities, and this will be done easily using R Generally speaking, I only want you to be familiar with the rule Such computations will be wrapped into other analyses later in the course and completely handled by a computer 19

20 Using R to Compute Normal Curve Percentages 1 We can use R as a calculator, i.e., an automatic standard normal table The relevant command that looks up values in a standard normal table is: For converting any standard deviation value (above or below the mean), z, to a corresponding proportion under a normal curve, the syntax is: pnorm(z) 20

21 Using R to Compute Normal Curve Percentages 2 As with any print version of a standard normal table, it is important to know what information pnorm(z) returns 21

22 Using R to Compute Normal Curve Percentages: Example Example: pnorm(1.0) 22

23 Summary The normal distribution is a theoretical probability distribution that is symmetric and bell-shaped There are literally an infinite number of normal distributions, and each can be completely specified by only two quantities: the mean and standard deviation For all normal distributions 68% of observations described by a normal distribution fall within 1 sd of the mean 95% of observations described by a normal distribution fall within 2 sds of the mean 99.7% of observations described by a normal distribution fall within 3 sds of the mean Other such percentages can be found using a standard normal table (available via R) 23

24 Applying the Principles of the Normal Distribution to Sample Data to Estimate Characteristics of Population Data The material in this video is subject to the copyright of the owners of the material and is being provided for educational purposes under rules of fair use for registered students in this course only. No additional copies of the copyrighted work may be made or distributed.

25 Learning Objectives Upon completion of this lecture, you will be able to: Create ranges containing a certain percentage of observations in an (approximately normal) distribution using only an estimate of the mean and standard deviation Figure out how far any individual data point is from the mean of its distribution in standardized units (compute a z-score) Convert z-scores to statements about relative proportions/probabilities for values that have an (approximately) normal distribution 2

26 The Normal Distribution, Generally Speaking 1 The normal distribution is a theoretical probability distribution No real data is perfectly described by this distribution For example, in a true normal distribution, the tails go on to negative and positive infinity, respectively 3

27 The Normal Distribution, Generally Speaking 2 However, the distributions of some data will be well approximated by a normal distribution In such situations we can use the properties of the normal curve to characterize aspects of the data distribution 4

28 Example: Histogram, 113 Systolic Blood Pressures 1 A basic histogram of these 113 measurements xx = mmhg ss = 12.9 mmhg mm = mmhg 5

29 Example: Histogram, 113 Systolic Blood Pressures 2 A basic histogram of these 113 measurements: normal distribution with same mean and standard deviation as sample superimposed xx = mmhg ss = 12.9 mmhg mm = mmhg 6

30 Using Sample Estimates and Properties of Normal Distribution to Estimate Percentiles Using only the sample mean and standard deviation, and assuming normality, let s estimate the 2.5 th and 97.5 th percentiles SBP in this population 2.5 th %ile: xx 2ss = (2 12.9) = 97.8 mmhg 97.5 th %ile: xx + 2ss = (2 12.9) = mmhg Based on this sample data, we estimate that most (95%) of the men in this clinical population have systolic blood pressures between 97.8 and mmhg Note: the observed 2.5 th and 97.5 th percentiles of the 113 sample value are mmhg and mmhg, respectively 7

31 Example: The z-score 1 Suppose you want to use the results from this sample of 113 men from a clinic to evaluate individual male patients relative to the population of all such patients For example, suppose a patient in your clinic has a SBP measurement of 130 mmhg. What proportion of men at the clinic have SBP measurements greater than this patient? 8

32 Example: The z-score 2 We want to figure out: xx = mmhg ss = 12.9 mmhg mm = mmhg 9

33 Example: The z-score 3 If we translate this measurement of 130 mmhg to units of standard deviation, we can find out how many sample standard deviations this person s SBP is above the sample mean. To do this: iiiiiiiiiiiiiiiiiiii oooooooooooooooooooooo Take xx ss 0.5 ssssssssssssssss dddddddddddddddddddd = = 6.4 mmhg 12.9 mmhg ss Now, the same question can be rephrased as, What percentage of observations in a normal curve are more than 0.5 SD above its mean? 10

34 Example: The z-score 4 What percentage of observations in a normal curve are greater than 0.5 SD above its mean? Using pnorm in R: 11

35 Example: The z-score 5 So 69% of the observations described by a standard normal curve are less than or equal to 0.5 standard deviations above the mean of 0 Hence, the remaining 31% are more than 0.5 standard deviations above 0 In terms of the original question posed, this means that an estimated 31% of the males in this population have blood pressures greater than 130 mmhg (i.e., using only the mean and sd, we have estimated the 69th percentile to be 130 mmhg) Just for context/comparison: the 70 th percentile of the observed 113 values is 130 mmhg 12

36 Example: The z-score 6 Another way to interpret this is as an (estimated) probability: the probability that any males in the population has a blood pressure measurement more than.5 standard deviations above the mean is.31 or 31% 13

37 The z-score, Generally Speaking 1 The type of computation we did to convert the SBP value of 130 to the number of SDs above (or below) the sample mean is sometimes called a z-score There is nothing special about a z-score; it is simply a measure of the relative distance (and direction) of a single observation in a data distribution relative to the mean of the distribution This distance is converted to units of standard deviation This is akin to converting kilometers to miles, or dollars to rupees 14

38 The z-score : A Parallel Example 1 You are an American who is apartment hunting in an unnamed European city. You wish to find an apartment within walking distance (+/- 1.5 miles) of the large organic supermarket, which is on Main Boulevard (E/W). You are only considering apartments on Main Blvd. The supermarket is 2 km west of the main city square. You are interested in 3 apartments: Apt 1 is 6 km west of the city square Apt 2 is.75 km west of the city square Apt 3 is 1 km east of the city square 15

39 The z-score : A Parallel Example 2 A schematic 16

40 The z-score : A Parallel Example 3 Apartment 1 is 4 km from the supermarket. This is the raw distance. We need to convert it to our standard units (miles) to interpret. 17

41 The z-score : A Parallel Example 4 How many miles is 4 km? Well, 1 mile = 1.6 km So 4 km = 4kkkk 2.5 miles east of the supermarket 1.6kkkk mmmmmmmm Similarly, Apartment 2 is 0.78 miles west of the supermarket (the conversion yields a value of -0.78), and Apartment 3 is miles west of the supermarket 18

42 The z-score, Generally Speaking 2 In some sense, the z-score is the statistical mile It allows to convert observations from different distributions with different measurement scales to comparable units When dealing with data that follow an (approximately) normal distribution, these z-scores tell us everything we need to know about the relative positioning of individual observations in the distribution of all observations We can compute z-scores for data arising from any type of distribution; however, for data from non-normal distributions, it will inform us about the relative position With non-normal data, this may not translate into correct percentile information 19

43 Example: Weight Data on Nepali 12-Month-Olds 1 A basic histogram of the weight for 236 Nepali children at one year old xx = 7.1 kg ss = 1.2 kg mm = 7.0 kg 20

44 Example: Weight Data on Nepali 12-Month-Olds 2 A basic histogram of the weight for 236 Nepali children at one year old, with normal curve with same mean and sd superimposed xx = 7.1 kg ss = 1.2 kg mm = 7.0 kg 21

45 Estimating Percentiles Based on Normality Assumption, Child Weight Data 1 Using only the sample mean and standard deviation, and assuming normality, let s estimate a range of weights for most (95%) Nepali children who were 12 months old 2.5 th %ile: xx 2ss = 7.1 (2 1.2) = 4.7 kg 97.5 th %ile: xx + 2ss = (2 1.2) = 9.5 kg Based on this sample data, we estimate that most (95%) of Nepali children who were 12 months had weights between 4.7 kg and 9.5 kg Note: the empirical 2.5 th and 97.5 th percentile of the 236 sample values are 4.4 kg and 9.7 kg, respectively 22

46 Relative Proportions Based on Normality Assumption, Child Weight Data 2 Suppose a mother brings her child to a pediatrician for the 12-month checkup, and wants to evaluate where the child s weight is relative to the population of 12-month-olds in Nepal Her child is 5 kg How does this child compare in weight to the weight of all 12-month-olds in Nepal? 23

47 Relative Proportions Based on Normality Assumption, Child Weight Data 3 The information we are trying to ascertain: 24

48 Relative Proportions Based on Normality Assumption, Child Weight Data 4 If we translate this measurement of 5 kg to units of standard deviation, we can find out where this child s weight compares to the mean of all such children Take iiiiiiiiiiiiiiiiiiii oooooooooooooooooooooo xx ss = = 2.1 kg 1.2 kg s 1.75 ssssssssssssssss dddddddddddddddddddd The original question ( How does my child s weight compare to the other children of the same age? ) can be asked as, What percentage of observations in a normal curve are more than 1.75 SDs below its mean? 25

49 Relative Proportions Based on Normality Assumption, Child Weight Data 5 The original question ( How does my child s weight compare to other children of the same age? ) can be asked as, What percentage of observations in a normal curve are more than 1.75 SDs below its mean? Using pnorm in R: 26

50 Relative Proportions Based on Normality Assumption, Child Weight Data 6 So 4% of the observations described by a standard normal curve are less than or equal to 1.75 standard deviations away from the mean of 0 In terms of the original question posed, this means that an estimated 4% of the children in this population have weights less than 5 kg (i.e., using only the mean and sd, we have estimated the 4 th percentile to be 5 kg) Just for context/comparison: the observed 5th percentile of these 236 measurements is 5 kg 27

51 Relative Proportions Based on Normality Assumption, Child Weight: A Broader Question 1 We could answer a broader question about the child who weighed 5 kg as well: What percentage of 12-month-old Nepali children have weights more extreme (unusual) than this child? What percentage of weights are farther than 1.75 SDs from the mean in either direction (above or below: i.e., z < or z > 1.75, sometimes expressed as z >1.75) Note: the above question can also be phrased, What is the probability that a 12-monthold Nepali child will have a weight measurement more than 1.75 SD from the mean of all such children (above or below)? 28

52 Relative Proportions Based on Normality Assumption, Child Weight: A Broader Question 2 Note: this question can also be phrased, What is the probability that a 12-month-old Nepali child will have a weight measurement more than 1.75 SD from the mean of all such children (above or below)? 29

53 Summary 1 The normal distribution is a theoretical probability distribution, which can be completely defined by two characteristics: the mean and standard deviation No real-world data has a perfect normal distribution; however, some continuous measures are reasonably approximated by a normal distribution 30

54 Summary 2 When dealing with samples from populations of (approximately) normally distributed data, the distribution of sample values will also be approximately normal. We can use the sample mean and standard deviation estimates, xx and ss, to: Create ranges containing a certain percentage of observations, or in other words: estimate the probability that an observed data point falls within a certain range of values Figure out how far any individual data point is from the mean of its distribution in standardized units (compute a z-score) Convert z-scores to statements about relative proportions/probabilities (and, hence, percentiles) for values that have an (approximately) normal distribution 31

55 What Happens When We Apply the Properties of the Normal Distribution to Data Not Approximately Normal?: A Warning The material in this video is subject to the copyright of the owners of the material and is being provided for educational purposes under rules of fair use for registered students in this course only. No additional copies of the copyrighted work may be made or distributed.

56 Learning Objectives Upon completion of this lecture, you will be able to: Describe situations in which using only the mean and standard deviation of a distribution of values to characterize the entire distribution will not work well Realize that z-scores are nothing special ; z-scores are just a (standardized) measure of distance Understand that z-scores do not necessarily align with the corresponding percentiles for a normal distribution for data that do not follow a normal distribution Choose the right approach to estimating ranges for individual values, and to computing percentage greater (or less) than a specific value using non-normal data distributions 2

57 The Theoretical Normal Distribution The normal distribution is a theoretical probability distribution No real data is perfectly described by this distribution For example, in a true normal distribution, the tails go on to negative and positive infinity, respectively 3

58 Applying the Normal Distribution Properties to Sample Data The distributions of some data will be well approximated by a normal distribution In such situations, we can use the properties of the normal curve to characterize aspects of the data distribution But the distributions of much data will not be well approximated by a normal distribution In such situations using the properties of the normal curve to characterize aspects of the data distribution will yield invalid results 4

59 Example: Histogram, Heritage Health Length of Stay Data 1 Histogram of the 12,928 length of stay values xx = 4.4 days ss = 4.7 days mm = 2.0 days Source: 5

60 Example: Histogram, Heritage Health Length of Stay Data 2 Histogram of the 12,928 length of stay values xx = 4.4 days ss = 4.7 days mm = 2.0 days Source: 6

61 Estimating Percentiles Based on Normality Assumption with Skewed Data (Length of Stay) 1 Using only the sample mean and standard deviation, and assuming normality, let s estimate the 2.5 th and 97.5 th percentiles for length of stay in this population 2.5 th %ile: xx 2ss = = 5.5 days 97.5 th %ile: xx + 2ss = = 14.1 days Based on this sample data, we estimate that most (95%) of the persons making claims in this health care population had length of stays between -5.5 and 14.1 days in 2011 (???????) Note: the empirical 2.5 th and 97.5 th percentile of the 12,298 sample values are 1 day and 20 days, respectively 7

62 Estimating Percentiles Based on Normality Assumption with Skewed Data (Length of Stay) 2 In this example, using the properties of the normal curve to estimate an interval containing the middle 95% of length of stay values for the claims population yields useless results Better to take the observed 2.5 th and 97.5 th percentiles of the sample data and report these as an estimate of the middle 95% Based on this sample data, we estimate that most (95%) of the persons making claims in this health care population had length of stays between 1 and 21 days in

63 Relative Proportions Based on Normality Assumption, Skewed Data (Length of Stay) 1 Suppose we wish to use these data to estimate the proportion of the claims population with total length of stay of greater than 5 days If we translate this measurement of 5 days to units of standard deviation, we can find where 5 days is relative to the sample mean length of stay. To do this, first find the zscore : Take iiiiiiiiiiiiiiiiiiii oooooooooooooooooooooo xx ss = = 0.7 days 0.14 ssssssssssssssss dddddddddddddddddddd 14.9 days s I ll let you verify that the probability of getting an observation that is greater than 0.14 SD above the mean of a normal distribution is 0.44 or 44%. 9

64 Relative Proportions Based on Normality Assumption, Skewed Data (Length of Stay) 2 However, if we look at some percentiles of the sample data: Percentile Value 10 th 1 day 25 th 1 day 50 th 2 days 60 th 4 days 70 th 4 days 75 th 5 days 80 th 6 days 90 th 10 days 97.5 th 20 days 10

65 Relative Proportions Based on Normality Assumption, Skewed Data (Length of Stay) 3 Based on these analyses, we estimate that approximately 25% of the claims had total length of stay greater than 5 days The above percentage (25%) is a lot smaller than the estimate of 44% we got using the mean and standard deviation to compute a z-score 11

66 Example: Histogram, CD4 Count Data CD4 counts for a random sample of 1,000 HIV-positive patients from a citywide clinical population xx = 280 cells/mm 3 ss = 198 cells/mm 3 mm = 249 cells/mm 3 Source: 12

67 Estimating Percentiles Based on Normality Assumption with Skewed Data (CD4 Counts) 1 Using only the sample mean and standard deviation, and assuming normality, let s estimate the 2.5 th and 97.5 th percentiles of CD4 counts in this population 2.5th %ile: 97.5th %ile: xx 2ss = 280 (2 198) = 116 cells/mm 3 xx + 2ss = (2 198) = 676 cells/mm 3 Based on this sample data, we estimate that most (95%) of population of HIV+ persons had CD4 counts between -116 and 676 cells/mm 3 Note: the empirical 2.5 th and 97.5 th percentile of the 1,000 sample value are 11 and 722 cells/mm 3 respectively 13

68 Summary While sample means and sample standard deviations are useful summary measures regardless of the data for which they are computed, these two quantities do not always help to characterize the data distribution (so far, this has worked only when the data distribution is approximately normal) For skewed distributions and others that are not approximately normal, using only the mean and standard deviation to characterize the entire underlying distribution can result in, at best, incorrect results and at worst, nonsensical results 14

69 Additional Examples The material in this video is subject to the copyright of the owners of the material and is being provided for educational purposes under rules of fair use for registered students in this course only. No additional copies of the copyrighted work may be made or distributed.

70 Example, Percentiles: Body Mass Index (BMI) 1 The 2015 Youth Risk Behavior Survey (YRBS) contains self-reported weight and height values for a large sample of US residents years in age These values can be used to compute body mass index (BMI) For this exercise, we will focus on the data for the 1, year-olds in the sample The mean BMI for this group is 23.6 kg/m 2, with standard deviation of 4.9 kg/m 2 These BMI values are approximately normally distributed Source: 2

71 Example, Percentiles: Body Mass Index (BMI) 2 In youth (< 18 years old), there is no singular cutoff for obesity The standard approach is to use the 95 th percentile for the BMI values for children of a given age Based on the information given on the previous slide, let s estimate the cutoff for obesity in the population of 16-year-olds 3

72 Example, Percentiles: Body Mass Index (BMI) 3 In order to do this using only the mean and standard deviation given, we need to know the value on a standard normal (i.e., number of standard deviations) that corresponds to the 95 th percentile The qnorm command in R will give this percentile The syntax is: qnorm(p), where p is the percentile of interest 4

73 Example, Percentiles: Body Mass Index (BMI) 4 For this example: 5

74 Example, Estimated Range: Body Mass Index (BMI) 1 In adults, BMIs between 18.5 and 24.9 are considered indicative of healthy weight Let s estimate the percentage of the 16-year-olds who have BMIs in this range 6

75 Example, Estimated Range: Body Mass Index (BMI) 2 First, some z-scores and a picture 7

76 Example, Estimated Range: Body Mass Index (BMI) 3 pnorm to the rescue! 8

77 Example: Baseline CD4 Counts By Treatment Response 1 Data from a random sample of 1,000 HIV-positive subjects xx = 292 cells/mm 3 ss = 194 cells/mm 3 mm = 257 cells/mm 3 xx = 234 cells/mm 3 ss = 208 cells/mm 3 mm = 196 cells/mm 3 9

78 Example: Baseline CD4 Counts By Treatment Response 2 To quantify the difference in distributions, we can use the mean difference xx nnnnnn rrrrrrrrrrrrrrrrrrrr xx rrrrrrrrrrrrrrrrrrrr = = 58 cells mm 3 xx rrrrrrrrrrrrrrrrrrrr xx nnnnnn rrrrrrrrrrrrrrrrrrrr = = 58 cells mm 3 Direction of comparison is arbitrary and will yield the same overall result, but it is important to specify and acknowledge so as to interpret correctly 10

79 Example: Baseline CD4 Counts By Treatment Response 3 Assuming (incorrectly) that the CD4 counts are normally distributed on both groups, let s figure out the range of values that covers the middle 95% of the data for the nonresponders 11

80 Example: Baseline CD4 Counts By Treatment Response 4 Using the observed 2.5 th and 97.5 th percentiles of these data gives an interval (for the middle 95%) of 14 cells/mm 3, 723 cells/mm 3 12