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Stats Lab

Math from your exam review · 20 questions per section · Fresh questions each set.

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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 − cfbelow) ÷ 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 = Xmax − XminDeviation = 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 − MX)(Y − MY)]

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

Pearson rStandardized cross products
r = SP ÷ √(SSX × SSY)

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

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.

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

Enter numbers only—no % signs or words. Round as requested in each question. Your score applies to the current set; New set generates 100 fresh questions.