Stats Lab · Practice Game

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Find the sum of squares (SS).

x = [2, 4, 6, 8]

Formula sheet · all calculations

Read left to right: what to find, the formula, then the step that matters. The symbols are defined below.

1 · Center and spread

Sample meanAverage of sample scores

M = ΣX ÷ n

Add every X; divide by the number of scores.

Population meanAverage of all population scores

μ = ΣX ÷ N

Same arithmetic; use N for the population count.

Weighted meanScores with frequencies

Σ(fX) ÷ Σf

Multiply each score by its frequency, add, then divide by total frequency.

Median and modeMiddle and most frequent

Median = middle ordered score; mode = most frequent score

For an even count, average the two middle scores.

Interpolated medianGrouped score intervals

L + [(n/2 − cf_below) ÷ f] × w

L is the median interval’s lower real limit; f is its frequency; w is interval width.

Range and deviationDistance across or from center

Range = X_max − X_minDeviation = X − M (or X − μ)

Sum of squaresAdd squared deviations

SS = Σ(X − M)²

For population data, use μ in place of M.

Sample variance and SDDenominator: n − 1

s² = SS ÷ (n − 1)s = √s²

Population variance and SDDenominator: N

σ² = SS ÷ Nσ = √σ²

Degrees of freedomSample

df = n − 1

2 · Frequencies, proportions, and percentiles

Proportion or probabilityPart of a total

p = f ÷ n

Count the desired outcomes and divide by the total.

PercentageOut of 100

Percentage = p × 100

Enter the numeric result without a % sign.

Cumulative frequencyAt or below X

cf = sum of frequencies through X

Cumulative percentage / percentile rankFrom a frequency table

(cf ÷ n) × 100

Use the cumulative frequency through the target score.

3 · Z-scores and raw scores

Population z-scoreRaw X to standardized z

z = (X − μ) ÷ σ

Subtract the population mean, then divide by population SD.

Sample z-scoreRaw X to standardized z

z = (X − M) ÷ s

Reverse: z to raw XUndo the standardization

X = μ + zσX = M + zs (sample)

Multiply z by the SD first, then add the mean.

4 · Normal table and probability

Body and tailUse the |z| row

Body + tail = 1

Body is the larger area; tail is the smaller area. For negative z, the tail is on the left.

Between mean and zColumn D in the exam table

Area = body − 0.5000

Left and right areasWhich side of z?

z > 0: left = body, right = tailz < 0: left = tail, right = body

Probability or percentile from XRaw score to area

X → z = (X − μ)/σ → left areaPercentile rank = left area × 100

X from a percentileArea back to raw score

Percentile ÷ 100 → table z → X = μ + zσ

Choose the sign of z from which side of the mean the percentile lies.

5 · Correlation

Sum of cross productsMultiply paired deviations

SP = Σ[(X − M_X)(Y − M_Y)]

Find each mean; multiply the deviations within each pair; add the products.

Pearson rStandardized cross products

r = SP ÷ √(SS_X × SS_Y)

Compute SS for X and Y separately. Then divide SP by the square root of their product.

6 · Sample means and the standard error

Standard error of MHow much sample means spread out

σ_M = σ ÷ √n

Bigger samples give a smaller standard error. The mean of the distribution of sample means is μ.

z-score for a sample meanLocate M in the distribution of sample means

z = (M − μ) ÷ σ_M

Find σ_M first, then use it in place of σ.

Sample size for a target standard errorWork the formula backwards

n = (σ ÷ σ_M)²

7 · Hypothesis tests, the t statistic, and effect size

Hypothesis test zKnown σ

z = (M − μ) ÷ σ_M, where σ_M = σ ÷ √n

Compare the result with the critical value (for example, ±1.96 for α = .05, two-tailed).

One-sample t statisticσ estimated from the sample

t = (M − μ) ÷ s_M, where s_M = s ÷ √n

Degrees of freedom: df = n − 1.

Cohen's dEffect size in standard-deviation units

d = (M − μ) ÷ σ (or s)

Roughly: 0.2 is small, 0.5 medium, 0.8 large.

Effect size r² for a t testPercentage of variance accounted for

r² = t² ÷ (t² + df)

8 · Two-sample t tests

Pooled varianceCombine two sample variances

s²_p = (SS₁ + SS₂) ÷ (df₁ + df₂)

Add the SS values and the df values separately, then divide.

Independent-measures tTwo separate samples

t = (M₁ − M₂) ÷ s_(M1−M2)s_(M1−M2) = √( s²_p÷n₁ + s²_p÷n₂ )

Degrees of freedom: df = df₁ + df₂.

Repeated-measures tDifference scores D

t = M_D ÷ s_MD, where s² = SS ÷ (n − 1) and s_MD = √(s² ÷ n)

Every calculation runs on the difference scores, not the two original samples.

9 · ANOVA

Sums of squaresThe pieces add up

SS_total = SS_between + SS_within

Degrees of freedom, one-wayk groups, N total scores

df_between = k − 1; df_within = N − k; df_total = N − 1

Mean squares and FVariance estimates and their ratio

MS = SS ÷ dfF = MS_between ÷ MS_within

Eta squaredEffect size for ANOVA

η² = SS_between ÷ SS_total

Degrees of freedom, two-factora levels of A, b levels of B, N total

df_A = a − 1; df_B = b − 1; df_A×B = (a − 1)(b − 1); df_within = N − (a × b); df_total = N − 1

10 · Regression and chi-square

Regression linePredict Y from X

b = SP ÷ SS_Xa = M_Y − b·M_XŶ = bX + a

Find the slope first; the intercept uses both means and the slope.

Chi-square goodness of fitOne variable, observed vs expected

χ² = Σ[ (f_o − f_e)² ÷ f_e ]

df = C − 1, where C is the number of categories. With equal proportions, f_e = N ÷ C.

Expected frequency, test for independenceOne cell of a two-way table

f_e = (row total × column total) ÷ N

df = (R − 1)(C − 1) for R rows and C columns.

Symbols

Σ = add all; X = raw score; f = frequency; n = sample count; N = population count; M = sample mean; μ = population mean; s = sample SD; σ = population SD; SS = sum of squared deviations; SP = sum of paired deviation products; cf = cumulative frequency; σ_M = standard error of M; s_M = estimated standard error; M_D = mean of the difference scores; k = number of groups; MS = mean square; F = MS ratio; η² = eta squared; f_o = observed frequency; f_e = expected frequency; χ² = chi-square statistic; Ŷ = predicted Y.

Open the standard normal table

This matches your exam table: body is the larger area, tail the smaller, and the last column is between the mean and z. Look up |z|; for negative z, the tail is on the left.

Standard normal areas by |z|
z