Chapter 3 (Summarizing Distributions)

Question 1

Mode

Answer 1

most likely value of a variable to occur

Question 2

Central tendency

Answer 2

values that are central in the distribution of a variable; describes what is typical

Question 3

Variation

Answer 3

describes how dispersed the data is over the range of possible values and what is atypical

Question 4

When a curve is bell-shaped (normal distribution), where does the mean, median, and mode lie?

Answer 4

They are all equal and lie in the middle of the distribution

Question 5

Sample (arithmetic) mean or average

Answer 5

most common measure of centrality; applies only to data where adding and dividing the values makes sense (nominal); has minimal variance (if replaced with any other number, variance would increase)

Question 6

Sample mean formula

Answer 6

the sum of all values of x from i=1 to n, divided by the number of observations or sample size (n)

Question 7

Weighted average

Answer 7

each observation gets a weight of 1/n, the proportion of the sample that it represents

Question 8

Weighted average

Answer 8

each observation gets a weight of 1/n, the proportion of the sample that it represents

Question 9

Dummy or binary variable

Answer 9

a qualitative variable that indicates the presence or absence of an attribute; must be coded as 1=present and 0=absent; also has a mean despite being qualitative

Question 10

Mean of a dummy variable

Answer 10

the proportion of the sample with the associated attribute

Question 11

How do you describe central tendency for qualitative variables?

Answer 11

(1) Create a dummy variable for each level of the qualitative variable (2) Summarize the mean of the dummy variable

Question 12

Percentiles

Answer 12

a way of describing how extreme a particular observation is (median is not extreme); the s-th percentile is the value of x such that s% of the data lies below it

Question 13

How do you get the median?

Answer 13

(1) Order x from smallest to largest (2) If n is odd, the median is the middle-most value. If n is even, the median is the average of the two middle-most values.

Question 14

Centrality of the median

Answer 14

the value that lies between two halves of all possible values

Question 15

Residual (ei)

Answer 15

a measure of variation; the difference between the proposed “typical” value (i.e. the sample mean) and the actual values

Question 16

Centrality of the sample mean

Answer 16

the sample mean is the value which is, on average, as close to the rest of the data as possible, and is subject to leverage by large or small values (i.e. outliers)

Question 17

How can you deal with outliers?

Answer 17

(1) Remove them from the dataset (2) Choose statistics that are robust to outliers like the median instead of the mean

Question 18

How is the residual a measure of variation?

Answer 18

it gives us a measure of how dispersed the value of xi is about the center xbar

Question 19

Bessel’s correction (n-1)

Answer 19

a statistical adjustment to make the sample variance and standard deviation more accurate or unbiased estimators of the population variance and standard deviation, particularly for small values of n

Question 20

Interquartile range (IQR)

Answer 20

IQR= x75 - x25; outlier robust because it is a percentile based measure like the median

Question 21

Sample standard deviation

Answer 21

square root of the sample variance

Question 22

Range

Answer 22

R= x100 - x0 = max(xi) - min(xi); not robust to outliers

Question 23

Percentile quintets

Answer 23

x0, x25, x50, x75, x100

Question 24

Covariance

Answer 24

Similar to variance but with two variables (instead of one) and has no equivalent of standard deviation

Question 25

Correlation or Pearson’s Correlation Coefficient (r)

Answer 25

a unitless statistic (just a number good for interpretation) that is always between -1 and 1; the covariance between x and y divided by the standard deviation of x and y

Question 26

Positive correlation

Answer 26

when r > 0, higher values of x result in higher values of y, and vice versa

Question 27

Negative correlation

Answer 27

when r < 0, higher values of x result in lower values of y, and vice versa

Question 28

No correlation

Answer 28

when r = 0, there is no relationship between values of x and y

Question 29

Perfect correlation

Answer 29

when r = +/- 1, the values of x and y can be perfectly predicted from one another

Question 30

How are correlation (r) and dependence related?

Answer 30

the r value is a numerical measurement of dependence in the data: close to -1 means strong negative dependence, close to +1 means strong positive dependence, close to 0 means a lack of dependence (“independent”)

Question 31

Population parameters

Answer 31

statistical objects that are sample analogues of important properties of the population distribution; population counterparts of the sample, as they have the same interpretation as the sample statistics

Question 32

Correspondences between sample statistics and population parameters

Answer 32

sample mean and population mean, sample variance and population variance, sample covariance and population covariance, sample correlation and population correlation

Question 33

Sampling distribution

Answer 33

shows every possible result a statistic can take in every possible (hypothetical) sample from a population and how often each result occurs; observations are unique samples and variables are statistics

Question 34

Empirical distribution

Answer 34

a very good estimate of the distribution in the population, gets closer if the data is representative and n is large

Question 35

Bootstrapping

Answer 35

simulating samples using the empirical distribution

Question 36

How do you implement the bootstrap?

Answer 36

(1) Randomly draw a new sample of the same size from the existing sample, with replacement (2) Do this hundreds or thousands of times to create a new sample of bootstrap samples (3) Compute the sampling distribution of your statistic from this sample

Question 37

Asymptotic behavior

Answer 37

how samples behave when n is large

Question 38

Asymptotic behavior of bootstrap

Answer 38

it gives us a sense of what the sampling distribution might look like and is centered around the sample statistics

Question 39

Law of Large Numbers (LLN)

Answer 39

if the sample is representative of the population (independent) and as n becomes large, the sample mean is a very close approximation of the population mean

Question 40

Central limit theorem (CLT)

Answer 40

given a population with a finite mean µ
and a finite non-zero variance σ2, the sampling distribution approaches a normal distribution with a variance of σ2/N, as N, the sample size, increases

Question 41

Useful properties of the normal or gaussian distribution

Answer 41

(1) symmetrical about the mean (2) the mean, median, and mode coincide (3) quantiles are closely related to the standard deviations (empirical rule): 68% of data within 1 sd, 95% within 2 sd, 99.7% within 3 sd

Question 42

Standard normal distribution

Answer 42

a normal distribution with a mean of 0 and a standard deviation of 1

Question 43

What is the difference between standard deviation and standard error (CLT)?

Answer 43

SD is the variation within a sample while SE is the variation between samples; SD is always bigger