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CHAPTER 2

CHAPTER 2. 2.1 - Basic Definitions and Properties Population Characteristics = “Parameters” Sample Characteristics = “Statistics” Random Variables ( Numerical vs. Categorical ) 2.2, 2.3 - Exploratory Data Analysis Graphical Displays Descriptive Statistics

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CHAPTER 2

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  1. CHAPTER 2 2.1 - Basic Definitions and Properties Population Characteristics = “Parameters” Sample Characteristics = “Statistics” Random Variables (Numerical vs. Categorical) 2.2, 2.3 - Exploratory Data Analysis Graphical Displays Descriptive Statistics Measures of Center (mode, median, mean) Measures of Spread (range, variance, standard deviation)

  2. “Classical Scientific Method” • Hypothesis–Define the study population... What’s the question? • Experiment– Designed to test hypothesis • Observations– Collect sample measurements • Analysis– Do the data formally tend to support or refute the hypothesis, and with what strength? (Lots of juicy formulas...) • Conclusion– Reject or retain hypothesis; is the result statistically significant? • Interpretation– Translate findings in context! Statistics is implemented in each step of the classical scientific method!

  3. Analysis– Do the data formally tend to support or refute the hypothesis, and with what strength? (Lots of juicy formulas...) • To help answer this question, we should first try to obtain an informal “feel” for the sample data we have collected, and see if it suggests anything about the population distribution. • ~ Exploratory Data Analysis ~ • Visual Displays (charts, tables, graphs, etc.) “What do the data look like?” • “Descriptive Statistics” (measures of center, measures of spread, proportions, etc.) “How can the data be summarized?”

  4. {18, 19, 19, 19, 20, 21, 21, 23, 24, 24, 26, 27, 31, 35, 35, 37, 38, 42, 46, 59} {18, 19, 19, 19, 20, 21, 21, 23, 24, 24, 26, 27, 31, 35, 35, 37, 38,42, 46,59} 4 values8 values5 values2 values 1 value From these values, we can construct a table which consists of the frequencies of each age-interval in the dataset, i.e., a frequency table. Frequency Histogram Example: Suppose the random variable isX = Age (years) in a certain population of individuals, and we select the following random sample of n = 20 ages. 8 Suggests population may be skewed to the right (i.e., positively skewed). “Endpoint convention” Here, the left endpoint is included, but not the right. Note!... Stay away from “10-20,” “20-30,” “30-40,” etc. 5 4 In published journal articles, the original data are almost never shown, but displayed in tabular form as above. This summary is called “grouped data.” 2 1

  5. Example: Suppose the random variable isX = Age (years) in a certain population of individuals, and we select the following random sample of n = 20 ages. {18, 19, 19, 19, 20, 21, 21, 23, 24, 24, 26, 27, 31, 35, 35, 37, 38,42, 46,59} Often though, it is preferable to work with proportions, i.e., relative frequencies… Divide frequencies by n = 20. ↓ Relative Frequency Histogram 0.4 0.3 0.2 0.1 0.0 .40 .25 .20 .10 .05 Relative frequencies are always between 0 and 1, and sum to 1.

  6. Example: Suppose the random variable isX = Age (years) in a certain population of individuals, and we select the following random sample of n = 20 ages. {18, 19, 19, 19, 20, 21, 21, 23, 24, 24, 26, 27, 31, 35, 35, 37, 38,42, 46,59} Often though, it is preferable to work with proportions, i.e., relative frequencies… Divide frequencies by n = 20. ↓ “0.00 of the sample is under 10 yrs old” Relative Frequency Histogram 0.4 0.3 0.2 0.1 0.0 “0.20 of the sample is under 20 yrs old” .40 “0.60 of the sample is under 30 yrs old” .25 “0.85 of the sample is under 40 yrs old” .20 “0.95 of the sample is under 50 yrs old” .10 .05 “1.00 of the sample is under 60 yrs old” Relative frequencies are always between 0 and 1, and sum to 1.

  7. Example: Suppose the random variable isX = Age (years) in a certain population of individuals, and we select the following random sample of n = 20 ages. {18, 19, 19, 19, 20, 21, 21, 23, 24, 24, 26, 27, 31, 35, 35, 37, 38,42, 46,59} Often though, it is preferable to work with proportions, i.e., relative frequencies… Divide frequencies by n = 20. ↓ Relative Frequency Histogram 0.4 0.3 0.2 0.1 0.0 .40 Cumulative relative frequencies always increase from 0 to 1. .25 .20 .10 .05 Relative frequencies are always between 0 and 1, and sum to 1. Approximately Example:Exactly what proportion of the sample is under 34 years old? [30, 34) contains 4/10 of 0.25 = 0.1, sum = 0.7 Solution: [0, 30) contains 0.6,

  8. Example: Suppose the random variable isX = Age (years) in a certain population of individuals, and we select the following random sample of n = 20 ages. {18, 19, 19, 19, 20, 21, 21, 23, 24, 24, 26, 27, 31, 35, 35, 37, 38,42, 46,59} Often though, it is preferable to work with proportions, i.e., relative frequencies… Divide frequencies by n = 20. ↓ Relative Frequency Histogram 0.4 0.3 0.2 0.1 0.0 .40 Cumulative relative frequencies always increase from 0 to 1. .25 .20 .10 .05 Relative frequencies are always between 0 and 1, and sum to 1. Example: Approximately what proportion of the sample is under 34 years old? Exactly But alas, there is a major problem…. [30, 34) contains 4/10 of 0.25 = 0.1, sum = 0.7 Solution: [0, 30) contains 0.6,

  9. Suppose that, for the purpose of the study, we are not primarily concerned with those 30 or older, and wish to “lump” them into a single class interval. {18, 19, 19, 19, 20, 21, 21, 23, 24, 24, 26, 27, 31, 35, 35, 37, 38,42, 46,59} 4 values8 values 8 values What effect will this have on the histogram? Relative Frequency Histogram .40 0.4 0.3 0.2 0.1 0.0 .40 .25 .20 .10 .05 The skew no longer appears. The histogram is distorted because of the presence of an outlier (59) in the data, creating the need for unequal class widths.

  10. (A Pain in the Tuches) Outliers What are they? Informally, an outlier is a sample data value that is either “much” smaller or larger than the other values. How do they arise? experimental error measurement error recording error not an error; genuine What can we do about them? double-check them if possible delete them? include them… somehow perform analysis both ways

  11. Density Histogram 0.04 0.02 0.40 0.0133… 0.20 0.40 IDEA: Instead of having height of each class rectangle = relative frequency, make... areaof each class rectangle = relative frequency. “Density” height × width = relative frequency / Total Area = 1! width = 10 width = 10 width = 30 The outlier is included, and the overall skewed appearance is restored. Exercise: What if the outlier was 99 instead of 59?

  12. Analysis– Do the data formally tend to support or refute the hypothesis, and with what strength? (Lots of juicy formulas...) • To help answer this question, we should first try to obtain an informal “feel” for the sample data we have collected, and see if it suggests anything about the population distribution. • ~ Exploratory Data Analysis ~ • Visual Displays (charts, tables, graphs, etc.)“What do the data look like?” • “Descriptive Statistics” (measures of center, measures of spread, proportions, etc.)“How can the data be summarized?”

  13. CHAPTER 2 2.1 - Basic Definitions and Properties Population Characteristics = “Parameters” Sample Characteristics = “Statistics” Random Variables (Numerical vs. Categorical) 2.2, 2.3 - Exploratory Data Analysis Graphical Displays Descriptive Statistics Measures of Center (mode, median, mean) Measures of Spread (range, variance, standard deviation)

  14. Center “Measures of ”Example: Sample exam scores = {70, 80, 80, 80, 80, 90, 90, 100, 100, 100} • sample mode most frequent value = 80 • sample median “middle” value = (80 + 90)/2 = 85 • sample mean average value = (Quartilesare found similarly: Q1 = , Q2 = 85, Q3 = ) 80 100 1/10 (70)(1) + (80)(4) + (90)(2) + (100)(3) = 87 xifi x =

  15. “Measures of Center”Example: Sample exam scores = {70, 80, 80, 80, 80, 90, 90, 100, 100, 100} • sample mode most frequent value = 80 • sample median “middle” value = (80 + 90)/2 = 85 • sample mean average value = 1/10 (70)(1) + (80)(4) + (90)(2) + (100)(3) = 87 xifi x =

  16. “Measures of Center”Example: Sample exam scores = {70, 80, 80, 80, 80, 90, 90, 100, 100, 100} = 87 x • sample mean 1/10 (70)(1) + (80)(4) + (90)(2) + (100)(3) = 87 1 10 4 10 2 10 3 10 1/10 (70)(1) + (80)(4) + (90)(2) + (100)(3) xif(xi) xifi “Notation, notation, notation.” x = x =

  17. “Measures of ”Example: Sample exam scores = {70, 80, 80, 80, 80, 90, 90, 100, 100, 100} Spread = 87 x … but how do we measure the “spread” of a set of values? • sample mean • First attempt: • sample range=xn – x1= 100 – 70 = 30. Simple, but… Ignores all of the data except the extreme points, thus far too sensitive to outliers to be of any practical value. Example: Company employee salaries, including CEO • Can modify with… • sample interquartile range (IQR) = Q3 – Q1 • = 100 – 80 = 20. We would still prefer a measure that uses all of the data.

  18. “Measures of ”Example: Sample exam scores = {70, 80, 80, 80, 80, 90, 90, 100, 100, 100} Spread = 87 x … but how do we measure the “spread” of a set of values? • sample mean Better attempt: Calculate the average of the “deviations from the mean.” 1/10 [(–17)(1) + (–7)(4) + (3)(2) + (13)(3)] = 0. ???????? This is not a coincidence – the deviations always sum to 0* – so it is not a good measure of variability. (xi– x) fi = * Physically, the sample mean is a “balance point” for the data.

  19. “Measures of Spread”Example: Sample exam scores = {70, 80, 80, 80, 80, 90, 90, 100, 100, 100} = 87 x • sample mean a modified average of the “squared deviations from the mean.” Calculate the [(–17)2(1) + (–7)2(4) + (3)2(2) + (13)2(3)] = 112.22 1/9 • sample variance  (xi– x)2fi s2 = • sample standard deviation s = s = 10.59

  20. Comments • is an unbiased estimator of the population mean , s2 is an unbiased estimator of the population variance  2. (Their “expected values” are  and  2, respectively.) • Beware of roundoff error!!! There is an alternate, more computationally stable formula for sample variance s2. • The numerator of s2 is called a sum of squares(SS); the denominator “n – 1” is the number of degrees of freedom(df) of the n deviations xi – , because they must satisfy a constraint (sum = 0), hence 1 degree of freedom is “lost.” • A natural setting for these formulas and concepts is geometric, specifically, the Pythagorean Theorem: a 2 + b 2 = c 2. See lecture notes appendix… c b a

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