Basic Statistics Formula Sheet 2n3q57

Basic Statistcs Formula Sheet Steven W. Nydick May 25, 2012

This document is only intended to review basic concepts/formulas from an introduction to statistics course. Only mean-based procedures are reviewed, and emphasis is placed on a simplistic understanding is placed on when to use any method. After reviewing and understanding this document, one should then learn about more complex procedures and methods in statistics. However, keep in mind the assumptions behind certain procedures, and know that statistical procedures are sometimes flexible to data that do not necessarily match the assumptions.

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Descriptive Statistics Elementary Descriptives (Univariate & Bivariate) Name

Population Symbol

Sample Symbol

Mean

µ

x ¯

Sample Calculation x ¯=

Main Problems

P

x N P

(x−¯ x)2 N −1

Sensitive to outliers

Median, Mode

Sensitive to outliers

MAD, IQR

Variance

σx2

s2x

s2x =

Standard Dev

σx

sx

sx =

Covariance

σxy

sxy

sxy =

P (x−¯ x)(y−¯ y) N −1

Outliers, uninterpretable units

Correlation

ρxy

rxy

rxy =

sxy sx sy

Range restriction, outliers,

rxy =

P (zx zy ) N −1

nonlinearity

z-score

zx

zx

zx =

√ 2 sx

x−¯ x ; sx

Biased

z¯ = 0; s2z = 1

Alternatives

MAD Correlation

Doesn’t make distribution normal

Simple Linear Regression (Usually Quantitative IV; Quantitative DV) Part Regular Equation

Population Symbol

Sample Symbol

Sample Calculation

Meaning

yi = α + βxi + i

yi = a + bxi + ei

yˆi = a + bxi

Predict y from x

sxy s2 x

Slope

β

b

b=

Intercept

α

a

a = y¯ − b¯ x

zyi = ρxy zxi + i

zyi = rxy zxi + ei

Standardized Equation Slope Intercept Effect Size

ρxy

rxy

None

None

P2

R2

=

P (x−¯ x)(y−¯ y) P (x−¯ x)2

Predicted y for x = 0

zˆyi = rxy zxi rxy =

2

sxy sx sy

Predicted change in y for unit change in x

=b

Predict zy from zx

sx sy



Predicted change in zy for unit change in zx

0

Predicted zy for zx = 0 is 0

2 ry2ˆy = rxy

Variance in y ed for by regression line

Inferential Statistics t-tests (Categorical IV (1 or 2 Groups); Quantitative DV) Test

Statistic x ¯

One Sample

¯ D

Paired Samples Independent Samples

x ¯1 − x ¯2

Parameter µ µD µ1 − µ2

Standard Deviation sx = sD = sp =

q

Standard Error

qP

(x−¯ x)2 N −1

sx √ N

qP

¯ 2 (D−D) ND −1

√sD

ND

2 (n1 −1)s2 1 +(n2 −1)s2 n1 +n2 −2

sp

q

1 n1

df

t-obt

N −1

tobt =

ND − 1

tobt =

¯ D−µ D0 s √D

tobt =

(¯ x1 −¯ x2 )−(µ1 −µ2 )0 q sp n1 + n1

x ¯−µ0 sx √ N

ND

+

1 n2

n1 + n2 − 2

1

r

ρ=0

a&b

α&β

Correlation Regression (FYI)

NA

N −2

tobt =

r r

sa & sb

N −2

tobt =

a−α0 sa

NA σ ˆe =

qP

(y−ˆ y )2 N −2

1−r 2 N −2

& tobt =

t-tests Hypotheses/Rejection Question

One Sample

Paired Sample

Independent Sample

Greater Than?

H0 : µ ≤ #

H0 : µD ≤ #

H 0 : µ1 − µ2 ≤ #

Extreme positive numbers

H1 : µ > #

H1 : µD > #

H 1 : µ1 − µ2 > #

tobt > tcrit (one-tailed)

H0 : µ ≥ #

H0 : µD ≥ #

H 0 : µ1 − µ2 ≥ #

Extreme negative numbers

H1 : µ < #

H1 : µD < #

H 1 : µ1 − µ2 < #

tobt < −tcrit (one-tailed)

H0 : µ = #

H0 : µD = #

H 0 : µ1 − µ2 = #

Extreme numbers (negative and positive)

H1 : µ 6= #

H1 : µD 6= #

H1 : µ1 − µ2 6= #

|tobt | > |tcrit | (two-tailed)

Less Than?

Not Equal To?

When to Reject

t-tests Miscellaneous Test One Sample Paired Samples Independent Samples

Confidence Interval: γ% = (1 − α)% x ¯ ± tN −1; crit(2-tailed) ×

Unstandardized Effect Size

sx √ N

x ¯ − µ0

¯ ± tN −1; crit(2-tailed) × √sD D D

¯ D

ND

(¯ x1 − x ¯2 ) ± tn1 +n2 −2; crit(2-tailed) × sp

q

1 n1

+

3

1 n2

x ¯1 − x ¯2

Standardized Effect Size dˆ =

x ¯−µ0 sx

dˆ = dˆ =

2

¯ D sD

x ¯1 −¯ x2 sp

b−β0 sb

One-Way ANOVA (Categorical IV (Usually 3 or More Groups); Quantitative DV) Source Between

Sums of Sq. Pg

j=1

nj (¯ xj − x ¯G )2

df

Mean Sq.

F -stat

g−1

SSB/df B

M SB/M SW

SSW/df W

Within

Pg

j=1 (nj

− 1)s2j

N −g

Total

P

i,j (xij

−x ¯G )2

N −1

Effect Size η2 =

SSB SST

1. We perform ANOVA because of family-wise error -- the probability of rejecting at least one true H0 during multiple tests. 2. G is “grand mean” or “average of all scores ignoring group hip.” 3. x ¯j is the mean of group j; nj is number of people in group j; g is the number of groups; N is the total number of “people”.

One-Way ANOVA Hypotheses/Rejection Question

Hypotheses

When to Reject

H 0 : µ1 = µ2 = · · · = µk

Is at least one mean different?

Extreme positive numbers

H1 : At least one µ is different from at least one other µ

Fobt > Fcrit

• Post-Hoc Tests: LSD, Bonferroni, Tukey (what are the rank orderings of the means?)

Chi Square (χ2 ) (Categorical IV; Categorical DV) Test Independence

Hypotheses H0 : Vars are Independent

Observed From Table

df

Expected N pj p k

(Cols - 1)(Rows - 1)

χ2 Stat PR PC i=1

j=1

When to Reject (fO ij −fE ij ) fE ij

H0 : Model Fits

From Table

N pi

Cells - 1

PC

i=1

(fO i −fE i )2 fE i

Extreme Positive Numbers χ2obt > χ2crit

H1 : Model Doesn’t Fit 1. 2. 3. 4.

Extreme Positive Numbers χ2obt > χ2crit

H1 : Vars are Dependent Goodness of Fit

2

: the sum is over the number of cells/columns/rows (not the number of people) For Test of Independence: pj and pk are the marginal proportions of variable j and variable k respectively For Goodness of Fit: pi is the expected proportion in cell i if the data fit the model N is the total number of people

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Assumptions of Statistical Models Correlation

Regression

1. Estimating: Relationship is linear

1. Relationship is linear

2. Estimating: No outliers

2. Bivariate normality

3. Estimating: No range restriction

3. Homoskedasticity (constant error variance)

4. Testing: Bivariate normality

4. Independence of pairs of observations

One Sample t-test

Independent Samples t-test

1. x is normally distributed in the population 2. Independence of observations

1. Each group is normally distributed in the population

Paired Samples t-test

2. Homogeneity of variance (both groups have the same variance in the population)

1. Difference scores are normally distributed in the population 2. Independence of pairs of observations

3. Independence of observations within and between groups (random sampling & random assignment)

One-Way ANOVA

Chi Square (χ2 )

1. Each group is normally distributed in the population

1. No small expected frequencies • Total number of observations at least 20 • Expected number in any cell at least 5

2. Homogeneity of variance

2. Independence of observations • Each individual is only in ONE cell of the table

3. Independence of observations within and between groups

Central Limit Theorem

Possible Decisions/Outcomes

H0 True H0 False Given a population distribution with a mean µ and a variance σ 2 , the sampling distribution of the mean using sample size N (or, to put it another way, the distribution Rejecting H0 Type I Error (α) Correct Decision (1 − β; Power) 2 of sample means) will have a mean of µx¯ = µ and a variance equal to σx2¯ = σN , Not Rejecting H0 Correct Decision (1 − α) Type II Error (β) which implies that σx¯ = √σN . Furthermore, the distribution will approach the normal 2 distribution as N , the sample size, increases. Power Increases If: N ↑, α ↑, σ ↓, Mean Difference ↑, or One-Tailed Test

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