week 4 -- SD as a ruler & z-scores

Question 1

Mean (equation)

Answer 1

mean y = sum of y / n

ȳ = sigma y / n

Question 2

descriptive vs. inferential

Answer 2

descriptive comes from our sample
inferential are statements about the population
–>
to make inferences about a population, I have to construct a model of that population

Question 3

Statistic

Answer 3

item of numerical info about the SAMPLE

Question 4

Paramater

Answer 4

item of numerical info about the MODEL (i.e., the POPULATION)

Question 5

Estimator

Answer 5

a statistic used to estimate a parameter (e.g., sample mean)

Question 6

Error

Answer 6

NON-SYSTEMATIC difference btw estimator and parameter

Question 7

Bias

Answer 7

SYSTEMATIC difference btw estimator and parameter

Question 8

Standard deviation – a measure to quantify spread of a sample (or population)

Allows us to answer: “How remarkable is a single observed value”?

algebraically = square root of variance
(square root of Σ (y- ȳ)2 / n
or with Bessel’s correction: Σ (y- ȳ)2 / n - 1

Answer 8

shows how close a data point is to the mean of the sample – BUT observations in a sample are always closer to their own mean than to the population mean. SO uncorrected SD is a biased estimator (OK as purely descriptive statistic)

Question 9

What is the trick for comparing performance btw very different-looking values (e.g., meters run vs. time ran)?

Answer 9

Standard deviation!
(use as a “ruler” to measure distance from the mean)

expressing distance with SD “standardizes” the performances

Question 10

z-score 1

allows us to compare apples and oranges (eliminates units)

Answer 10

letter z denotes values that have been standardized!!
(with mean & SD)

z = y - ȳ / s

z-score = performance - mean performance / standard deviation

Question 11

z-score 2

Comparsion shows us which score is more extraordinary

Answer 11

z-scores have NO UNITS
they tell us how far the data is from the mean
2 = 2 SD above the mean
-1.5 = 1.5 SD below the mean

Question 12

shifting data

plus or minus

Answer 12

Only measures of position change (center, min, max)

Neither shape nor spread changes (range, IQR, SD)

Question 13

rescaling data

multiply or divide

Answer 13

all measures of position (mean, median) and spread change

shape remains constant

Question 14

standardizing into z-scores shifts data by the mean and rescales them by the standard deviation

Answer 14

Shape stays constant
center changes (mean = 0)
spread changes (SD = 1)

Question 15

A statistical model is always wrong. Explain.

Answer 15

it is “wrong” in the sense that it doesn’t match reality exactly

Question 16

Normal model

Answer 16

a way to show how extreme a z-score is

Question 17

N (μ, σ)

(Normal model

Answer 17

mew μ, sigma σ
μ = mean of a Normal model
σ = standard deviation of a Normal model

Question 18

Greek letters

Answer 18

NOT numerical summaries of data – they are part of the model, parameters

Question 19

Latin letters

Answer 19

summaries of data, statistics

Question 20

standardized data for model

Answer 20

z = y - μ / σ (for parameters)

cf.
z = y - ȳ / s (for statistics)

Question 21

68 - 95 - 99,7 rule

Answer 21

In a Normal model:
68% of values fall within 1 SD of the mean
95% of values fall within 2 SD of the mean
99,7% of values fall within 3 SD of the mean