Single Sample t-Test

Descriptive Statistics Formula Sheet

Sample Population

Characteristic statistic Parameter

raw scores x, y, . . . . . X, Y, . . . . .

mean (central tendency) M = ∑ x

n μ =

∑ X

N

range (interval/ratio data) highest minus lowest value highest minus lowest value

deviation (distance from mean) Deviation = (x − M ) Deviation = (X − μ )

average deviation (average distance from mean)

∑(x − M )

n = 0

∑(X − μ )

N

sum of the squares (SS) (computational formula) SS = ∑ x

2 − (∑ x)2

n SS = ∑ X2 −

(∑ X)2

N

variance ( average deviation2 or standard deviation

2 )

(computational formula) s2 =

∑ x2 − (∑ x)2

n n − 1

= SS

df σ2 =

∑ X2 − (∑ X)2

N N

standard deviation (average deviation or distance from mean) (computational formula) s =

√∑ x 2 −

(∑ x)2

n n − 1

σ = √∑ X

2 − (∑ X)2

N N

Z scores (standard scores)

mean = 0 standard deviation = ± 1.0

Z = x − M

s =

deviation

stand. dev.

X = M + Zs

Z = X − μ

σ

X = μ + Zσ

Area Under the Normal Curve -1s to +1s = 68.3% -2s to +2s = 95.4% -3s to +3s = 99.7%

Using Z Score Table for Normal Distribution (Note: see graph and table in A-23)

for percentiles (proportion or %) below X for positive Z scores – use body column for negative Z scores – use tail column for proportions or percentage above X for positive Z scores – use tail column for negative Z scores – use body column to discover percentage / proportion between two X values

1. Convert each X to Z score 2. Find appropriate area (body or tail) for each Z score 3. Subtract or add areas as appropriate 4. Change area to % (area × 100 = %)

Regression lines (central tendency line for all points; used for predictions only) formula uses raw scores b = slope a = y-intercept

y = bx + a (plug in x to predict y)

b = ∑ xy −

(∑ x)(∑ y) n

∑ x2 − (∑ x)2

n

a = My - bMx where My is mean of y and Mx is mean of x

SEest (measures accuracy of predictions; same properties as standard deviation)

Pearson Correlation Coefficient (used to measure relationship; uses Z scores)

r = ∑ xy−

(∑ x)(∑ y)

n

√(∑ x2− (∑ x)2

n )(∑ y2−

(∑ y)2

n )

r = degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟

degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑒𝑙𝑦

r

2 = estimate or % of accuracy of predictions