Descriptive statistics
- Mean
- x̄ = Σx/n
- Sample standard deviation
- s = √(Σ(x − x̄)²/(n − 1))
- Median
- middle value of the sorted data
- z-score
- z = (x − μ)/σ
- Normal distribution
- 68% within ±1σ, 95% within ±2σ
Introductory Statistics: cram pack
Formula sheets, the key method steps for every topic and 12 practice problems, with full solutions at the back.
1Formulas and methods
Probability
- Complement
- P(not A) = 1 − P(A)
- Addition rule
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Independent events
- P(A ∩ B) = P(A)·P(B)
- Conditional probability
- P(A|B) = P(A ∩ B)/P(B)
- Binomial
- P(X = k) = C(n,k)·pk(1 − p)n−k
Normal distribution
- z-score
- z = (x − μ)/σ
- Below x
- P(X < x) = Φ(z)
- Above x
- P(X > x) = 1 − Φ(z)
- Between
- Φ(z2) − Φ(z1)
- Percentile
- x = μ + z·σ
Method
- Convert the value to a z-score: z = (x − μ)/σ.
- Look up Φ(z), the area to the left of z.
- For the area to the right use 1 − Φ(z); for an interval subtract the two left areas.
- For a percentile, work backwards: find z for the area, then x = μ + z·σ.
Confidence intervals
- σ known
- x̄ ± z·σ/√n
- σ unknown
- x̄ ± t·s/√n, df = n − 1
- Critical z
- 90%: 1.645, 95%: 1.96, 99%: 2.576
- Sample size
- n = (z·σ/E)², rounded up
Method
- Decide between z (population σ known) and t (only the sample s, with n − 1 degrees of freedom).
- Find the critical value for the confidence level.
- Margin of error: E = critical value × standard deviation/√n.
- Interval: x̄ ± E.
Hypothesis tests & confidence intervals
- z-test for a mean
- z = (x̄ − μ0)/(σ/√n)
- t-test for a mean
- t = (x̄ − μ0)/(s/√n), df = n − 1
- Two-sided at 5%
- reject H0 if |z| > 1.96
- Confidence interval
- x̄ ± z·σ/√n
Method
- State H₀ and H₁ and decide whether the test is one- or two-sided.
- Calculate the test statistic: (x̄ − μ₀)/(σ/√n), or with s and t.
- Compare with the critical value, or find the p-value.
- Reject H₀ if the statistic is beyond the critical value (p below α), and state the conclusion in words.
Linear regression & correlation
- Sums of squares
- Sxx = Σ(x − x̄)² = Σx² − (Σx)²/n, Sxy = Σxy − ΣxΣy/n
- Slope
- b = Sxy/Sxx
- Intercept
- a = ȳ − b·x̄
- Correlation
- r = Sxy/√(SxxSyy)
- Residual
- e = y − ŷ
2Practice problems
1.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 10, 30, 22, 35, 16, 20.
n = 6
s =
2.Descriptive statisticsHard
Exam scores are normally distributed with mean 500 and standard deviation 100. Find the z-score z of a score of 300.
μ = 500σ = 100x = 300
z =
3.ProbabilityMedium
A bag holds 3 red and 3 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?
3 / 3
P =
4.ProbabilityHard
A test has 5 multiple-choice questions. A student who guesses gets each one right with probability 0.1, independently. What is the probability P of exactly 4 correct answers?
n = 5p = 0.1k = 4
P =
5.Normal distributionMedium
Drag z so that the shaded area between −z and z under the standard normal curve is 0.8557. Find z to two decimals.
A = 0.8557
z =
6.Normal distributionHard
Hole diameters D are normal with mean 25.05 mm and standard deviation 0.015 mm. Shaft diameters d are independent and normal with mean 25 mm and standard deviation 0.015 mm. A randomly chosen shaft fits when D − d > 0. Find the probability P that it fits. Round z to two decimals.
hole: μ = 25.05 mm, σ = 0.015 mmshaft: μ = 25 mm, σ = 0.015 mm
P =
7.Confidence intervalsMedium
A 90% confidence interval for a mean, based on 100 observations and a known σ, has a margin of error of 4. What is the margin of error E₂ of a 95% interval from the same data?
90%n₁ = 100E₁ = 495%
E₂ =
8.Confidence intervalsHard
The population standard deviation is 35. How large a sample n do you need so that a 90% confidence interval for the mean has a margin of error of at most 2.5?
σ = 35E = 2.590%
n =
9.Hypothesis tests & confidence intervalsMedium
13 employees take a typing test before and after a training course. For each person the difference d = after − before (words per minute) is recorded. The mean difference is −3.75 and the standard deviation of the differences is 4.1. Find the paired test statistic t for H₀: μd = 0.
n = 13d̄ = −3.75sd = 4.1
t =
10.Hypothesis tests & confidence intervalsHard
A one-sided z test at α = 0.01 must detect a shift of the mean of 4 units with a power of 95%. The standard deviation is 14. Find the minimum sample size n.
α = 0.01power = 95%δ = 4σ = 14zα = 2.326zβ = 1.645
n =
11.Linear regression & correlationMedium
The table gives 6 paired observations of x = temperature in °C and y = ice creams sold. Find the coefficient of determination r² of the least-squares line.
n = 6
r² =
12.Linear regression & correlationHard
The table gives 5 paired observations of x = advertising spend in k€ and y = sales in k€. Use the least-squares line ŷ = a + b·x. A new observation (2.5, 47) is made. Find its residual e = y − ŷ.
n = 5(2.5, 47)
e =
3Solutions
1.Descriptive statistics
Answer: s = 9.131
Descriptive statistics: the mean is the sum divided by n; the median is the middle value of the sorted data (the average of the two middle values when n is even). The sample standard deviation is s = √(Σ(x − x̄)²/(n − 1)). A z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean.
Mean
x̄ = 133/6 = 22.17
Squared deviations
Σ(x − x̄)² = (−12.17)² + 7.833² + (−0.1667)² + 12.83² + (−6.167)² + (−2.167)² = 416.8
Standard deviation
s = √(416.8/5) = 9.131
Answer
s = 9.131
2.Descriptive statistics
Answer: z = −2
Descriptive statistics: the mean is the sum divided by n; the median is the middle value of the sorted data (the average of the two middle values when n is even). The sample standard deviation is s = √(Σ(x − x̄)²/(n − 1)). A z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean.
Standardise
z = (300 − 500)/100 = −2
Answer
z = −2
3.Probability
Answer: P = 0.2
Probability: with equally likely outcomes, P(A) = favourable/total. For independent events multiply: P(A and B) = P(A)·P(B). The complement often helps: P(at least one) = 1 − P(none). Binomial: in n independent trials with success probability p, P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ.
First and second draw
P = 3/6 · 2/5 = 6/30 = 0.2
Answer
P = 0.2
4.Probability
Answer: P = 0.00045
Probability: with equally likely outcomes, P(A) = favourable/total. For independent events multiply: P(A and B) = P(A)·P(B). The complement often helps: P(at least one) = 1 − P(none). Binomial: in n independent trials with success probability p, P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ.
Binomial formula
P(X = 4) = 5·0.14·0.91 = 5·0.0001·0.9 = 0.00045
Answer
P = 0.00045
5.Normal distribution
Answer: z = 1.46
From an area back to z: Φ(z) is the area to the left of z. For a left-tail area A, find z with Φ(z) = A in the table. A right-tail area A means Φ(z) = 1 − A. A central area A between −z and z leaves (1 − A)/2 in each tail, so Φ(z) = (1 + A)/2. For a value x, use x = μ + zσ.
Area to the left of z
Φ(z) = (1 + 0.8557)/2 = 0.9279
Read the z-table backwards
Find 0.9279 inside the table and read off z.
z = 1.46
Answer
z = 1.46
6.Normal distribution
Answer: P = 0.9909
Sums and differences of normal variables: if X and Y are independent and normal, X + Y and X − Y are normal too. Means add or subtract: E(X ± Y) = μX ± μY. Variances always add, also for a difference: Var(X ± Y) = σX² + σY², so σ = √(σX² + σY²). Standard deviations never add directly. The total of n independent items with the same distribution has mean n·μ and standard deviation σ·√n. Then standardise as usual: z = (x − mean)/σ and read Φ(z).
Distribution of the difference
μ = 25.05 − 25 = 0.05 mm, σ = 0.02121 mm
Standardise
z = (0 − 0.05)/0.02121 = −2.36
Probability
P = 1 − Φ(−2.36) = 0.9909
Answer
P = 0.9909
7.Confidence intervals
Answer: E₂ = 4.766
Confidence interval for a mean: x̄ ± margin of error. With a known population σ the margin is z·σ/√n; with only the sample standard deviation s, use t with n − 1 degrees of freedom: t·s/√n. Critical values: z = 1.645 (90%), 1.96 (95%), 2.576 (99%). To get a margin E or smaller, you need n ≥ (z·σ/E)², rounded up.
Ratio of margins
E₂/E₁ = (1.96/1.645)·√(100/100) = 1.191
New margin
E₂ = 4·1.191 = 4.766
Answer
E₂ = 4.766
8.Confidence intervals
Answer: n = 531
Confidence interval for a mean: x̄ ± margin of error. With a known population σ the margin is z·σ/√n; with only the sample standard deviation s, use t with n − 1 degrees of freedom: t·s/√n. Critical values: z = 1.645 (90%), 1.96 (95%), 2.576 (99%). To get a margin E or smaller, you need n ≥ (z·σ/E)², rounded up.
Solve for n
n = (1.645·35/2.5)² = 530.4
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 531
Answer
n = 531
9.Hypothesis tests & confidence intervals
Answer: t = −3.298
Testing a mean: the test statistic measures how many standard errors the sample mean lies from the value claimed in H₀: z = (x̄ − μ₀)/(σ/√n) when σ is known, t = (x̄ − μ₀)/(s/√n) when it is estimated from the sample. A confidence interval is x̄ ± (critical value)·(standard error).
Standard error of the mean difference
SE = 4.1/√13 = 1.137
Test statistic
t = −3.75/1.137 = −3.298
Answer
t = −3.298
With df = 12, the two-sided 5% critical value is 2.179; |t| is larger, so H₀ is rejected.
10.Hypothesis tests & confidence intervals
Answer: n = 194
Sample size for a test: to detect a true shift δ with power 1 − β in a one-sided z test at level α, the standard error must satisfy δ = (z_α + z_β)·σ/√n, so n = ((z_α + z_β)·σ/δ)², rounded up.
Solve for n
n = ((2.326 + 1.645)·14/4)² = 193.2
Round up
n = 194
Answer
n = 194
11.Linear regression & correlation
Answer: r² = 0.9666
Least-squares regression: with x̄ and ȳ the means, Sxx = Σ(x − x̄)², Syy = Σ(y − ȳ)² and Sxy = Σ(x − x̄)(y − ȳ), the line ŷ = a + b·x has slope b = Sxy/Sxx and intercept a = ȳ − b·x̄, so it always passes through (x̄, ȳ). Pearson's correlation coefficient r = Sxy/√(Sxx·Syy) lies between −1 and 1; r² is the fraction of the variation in y explained by the line.
Means
x̄ = 153/6 = 25.5
ȳ = 465/6 = 77.5
Sums of squares and products
Sxx = Σ(x − x̄)² = 157.5
Sxy = Σ(x − x̄)(y − ȳ) = 586.5
Syy = Σ(y − ȳ)² = 2260
Coefficient of determination
r² = 586.5²/(157.5·2260) = 0.9666
Answer
r² = 0.9666
About 96.66% of the variation in y is explained by the line.
12.Linear regression & correlation
Answer: e = −3.65
Least-squares regression: with x̄ and ȳ the means, Sxx = Σ(x − x̄)², Syy = Σ(y − ȳ)² and Sxy = Σ(x − x̄)(y − ȳ), the line ŷ = a + b·x has slope b = Sxy/Sxx and intercept a = ȳ − b·x̄, so it always passes through (x̄, ȳ). Pearson's correlation coefficient r = Sxy/√(Sxx·Syy) lies between −1 and 1; r² is the fraction of the variation in y explained by the line.
Means
x̄ = 15/5 = 3
ȳ = 266/5 = 53.2
Sums of squares and products
Sxx = Σ(x − x̄)² = 10
Sxy = Σ(x − x̄)(y − ȳ) = 51
Regression line
b = 51/10 = 5.1
a = 53.2 − 5.1·3 = 37.9
ŷ = 37.9 + 5.1·x
Predicted value
ŷ = 37.9 + 5.1·2.5 = 50.65
Residual
e = 47 − 50.65 = −3.65
Answer
e = −3.65
The point lies below the line.
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