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Formula sheet for final exam fall semester...

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SDS 302 Summary of Statistics Formulas Mean

𝑥 =

∑𝑥 𝑛

Sample Standard Deviation

∑(𝑥 − 𝑥 )2 𝑠= √ 𝑛−1 s2 = sample variance

x = data value n = sample size

Alternatively written as: 𝑠= √

SS stands for “sum of squares”

𝑆𝑆

→ 𝑆𝑆 = ∑(𝑥 − 𝑥 )2 or 𝑆𝑆 = ∑ 𝑥 2 −

𝑛−1

(∑ 𝑥)

2

𝑛 Hand-calculation shortcut

Definitional formula

Interquartile Range IQR = Q3 - Q1

z-score 𝑧=

𝑥 − 𝑥 𝑠

Pearson Correlation ∑ 𝑧𝑥 𝑧𝑦 𝑟= 𝑛−1

Q1 is the median of the lower half of the sorted data. Q3 is the median of the upper half of the sorted data. (When finding Q1 and Q3, do not include the median if the data has an odd number of observations)

When working from a model, z-score 𝑧=

𝑥−𝜇 𝜎

µ = model mean, σ = model standard deviation

For a given data point: 𝑧𝑥 =

𝑥−𝑥 𝑠

and 𝑧𝑦 =

𝑦−𝑦 𝑠

Chi Square 𝑋2 = ∑

(𝑂𝑏𝑠 − 𝐸𝑥𝑝)2 𝐸𝑥𝑝

Linear Regression 𝑦 = 𝑏0 + 𝑏1 𝑥 Exponential Model 𝑓(𝑡) =

𝑎𝑏𝑡

Independence:

Goodness of Fit:

df = (# rows-1)(# columns-1) 𝑏1 = 𝑟 b=1+r

𝑠𝑦 𝑠𝑥

𝑏0 = 𝑦 − 𝑏1 𝑥

df = # of categories – 1

(r = correlation)

(r = growth rate, expressed as decimal)

y= log x means 10 y = x

Converting between base e and base b: 𝑎𝑒 𝑘𝑡 = 𝑎𝑏𝑡

Formulas for Making Statistical Inferences from Samples Sampling Distribution Model Mean

One-sample t-test Independent samples t-test

Paired samples t-test

Estimate of Standard Error 𝑠𝑥 =

µ 𝜇1 − 𝜇2

𝑠

√𝑛

Variance not equal:

𝑠𝑥1−𝑥 2 = √

𝜇𝑑

𝑠12 𝑠22 + 𝑛1 𝑛2

𝑠𝑑 = 𝑠𝑑 /√𝑛

Source

ANOVA

Between/Factor/Explained Within/Error/Unexplained Total Effect size for ANOVA: 𝜂2 =

𝑆𝑆𝐵 𝑆𝑆𝑇

Confidence Interval

Observed ± margin of error 𝑠

𝑥 ± 𝑡𝑐

Hypothesis Test Statistic 𝑡=

(𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛) − (ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑧𝑒𝑑 𝑚𝑒𝑎𝑛) 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟

𝑑 ± 𝑡𝑐

𝑠12 𝑠22 + 𝑛1 𝑛2

𝑡=

𝑠𝑑

∑ ∑(𝑥𝑗 − 𝑥 )2

∑ ∑(𝑥𝑖𝑗 − 𝑥𝑗 )2 ∑ ∑(𝑥𝑖𝑗 − 𝑥 )2

n-1

(𝑥1 − 𝑥2 ) − (µ1 − µ2 ) 𝑠𝑥1−𝑥 2

Smaller of 𝑛1 − 1 and 𝑛2 − 1

𝑑 − µ𝑑

n-1

𝑡=

√𝑛

SS

𝑥 − µ

𝑡=

√𝑛

𝑥1 − 𝑥2 ± 𝑡𝑐 √

Degrees of Freedom

𝑠/√𝑛

𝑠𝑑 /√𝑛

df

MS

F-Statistic

k-1

SSB/(k-1)

MSB/MSW

N-k

SSW/(N-k)

N-1

Post-hoc: Bonferroni adjustment 1. Perform an independent t-test for each pair of group means 2. Find the p-value associated with each test 3. Compare the p-value with a significance level of /m where m is the number of independent t-tests performed

Standard Normal Cumulative Probabilities

Cumulative probability for z is the area under the standard normal curve to the left of z.

(Cumulative probability)

0

Standard Normal Cumulative Probabilities (continued) Cumulative probability for z is the area under the standard normal curve to the left of z.

(Cumulative probability)

0...