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: 49, 39, 50, 10, 29, 40, 44.
n = 7
s =
2.Descriptive statisticsHard
Exam scores are normally distributed with mean 50 and standard deviation 10. Find the z-score z of a score of 56.
μ = 50σ = 10x = 56
z =
3.ProbabilityMedium
For the distribution shown, find the variance Var(X) of X.
x: 0, 2, 4, 6p(x): 0.25, 0.3, 0.3, 0.15
Var(X) =
4.ProbabilityHard
A test has 9 multiple-choice questions. A student who guesses gets each one right with probability 0.4, independently. What is the probability P of exactly 2 correct answers?
n = 9p = 0.4k = 2
P =
5.Normal distributionMedium
The masses of bags of cement are independent and normal with mean 25 kg and standard deviation 0.5 kg. Find the probability P that the total mass of 10 of them exceeds 250.8 kg. Round z to two decimals.
μ = 25 kgσ = 0.5 kgn = 10> 250.8 kg
P =
6.Normal distributionHard
The exam scores are normally distributed with mean 65 points and standard deviation 10 points. Find the value x below which 99% of them lie (the 99th percentile).
μ = 65σ = 10P = 99%
x = points
7.Confidence intervalsMedium
A sample of 30 items has mean 30.5 and standard deviation s = 13. The population is roughly normal. Find the upper limit of the 95% confidence interval for the mean.
n = 30x̄ = 30.5s = 1395%t(29) = 2.045
upper limit =
8.Confidence intervalsHard
The population standard deviation is 35. How large a sample n do you need so that a 95% confidence interval for the mean has a margin of error of at most 3?
σ = 35E = 395%
n =
9.Hypothesis tests & confidence intervalsMedium
The mean of a population is claimed to be 500; the population standard deviation is known to be 5. A random sample of 100 observations has a mean of 499.1. Test H₀: μ = 500 against H₁: μ ≠ 500. Find the p-value p.
μ₀ = 500σ = 5n = 100x̄ = 499.1H₁: μ ≠ μ₀
p =
10.Hypothesis tests & confidence intervalsHard
In a trial, 165 of 300 patients on treatment A recover, and 83 of 120 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
x₁ = 165n₁ = 300x₂ = 83n₂ = 120
z =
11.Linear regression & correlationMedium
The table gives 5 paired observations of x = load in kN and y = extension in mm. Find Pearson's correlation coefficient r.
n = 5
r =
12.Linear regression & correlationHard
The table gives 5 paired observations of x = temperature in °C and y = ice creams sold. Use the least-squares line ŷ = a + b·x. A new observation (22.5, 70) is made. Find its residual e = y − ŷ.
n = 5(22.5, 70)
e =
3Solutions
1.Descriptive statistics
Answer: s = 13.95
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 = 261/7 = 37.29
Squared deviations
Σ(x − x̄)² = 11.71² + 1.714² + 12.71² + (−27.29)² + (−8.286)² + 2.714² + 6.714² = 1167
Standard deviation
s = √(1167/6) = 13.95
Answer
s = 13.95
2.Descriptive statistics
Answer: z = 0.6
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 = (56 − 50)/10 = 0.6
Answer
z = 0.6
3.Probability
Answer: Var(X) = 4.11
Discrete random variables: a probability distribution lists every value x a variable can take with its probability p(x); the probabilities are between 0 and 1 and add up to 1. The expected value (mean) is E(X) = Σx·p(x), the long-run average. The variance is Var(X) = Σ(x − μ)²·p(x) = E(X²) − μ², and the standard deviation σ = √Var(X). For a linear change Y = aX + b: E(Y) = a·E(X) + b, Var(Y) = a²·Var(X) and σY = |a|·σX; adding a constant does not change the spread.
Mean
μ = 0·0.25 + 2·0.3 + 4·0.3 + 6·0.15 = 2.7
Mean of the squares
E(X²) = 0·0.25 + 4·0.3 + 16·0.3 + 36·0.15 = 11.4
Variance
Var(X) = 11.4 − 2.7² = 4.11
Answer
Var(X) = 4.11
4.Probability
Answer: P = 0.1612
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 = 2) = 36·0.42·0.67 = 36·0.16·0.02799 = 0.1612
Answer
P = 0.1612
5.Normal distribution
Answer: P = 0.305
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 total
μT = 250 kg, σT = 0.5·√10 = 1.581 kg
Standardise
z = (250.8 − 250)/1.581 = 0.51
Probability
P = 1 − Φ(0.51) = 0.305
Answer
P = 0.305
6.Normal distribution
Answer: x = 88.26 points
Normal distribution: convert a value x to a z-score, z = (x − μ)/σ, which counts standard deviations from the mean. A z-table (or Φ(z)) gives the area to the left of z, which is P(X < x). The area to the right is 1 − Φ(z), and the area between two values is the difference of their two areas.
z for the percentile
Φ(z) = 0.99 → z = 2.326
Back to x
x = 65 + 2.326·10 = 88.26 points
Answer
x = 88.26 points
7.Confidence intervals
Answer: upper limit = 35.35
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.
Critical value
df = n − 1 = 29, t = 2.045
Margin of error
E = 2.045·13/√30 = 4.854
Interval
30.5 ± 4.854 → [25.65, 35.35]
Answer
upper limit = 35.35
8.Confidence intervals
Answer: n = 523
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 = (1.96·35/3)² = 522.9
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 523
Answer
n = 523
9.Hypothesis tests & confidence intervals
Answer: p = 0.07186
Hypothesis test: the test statistic = (estimate − value under H₀)/(standard error under H₀). The p-value is the probability, assuming H₀ is true, of a statistic at least as extreme as the one observed: the area in one tail for a one-sided H₁, twice the tail area for a two-sided H₁. Reject H₀ when p ≤ α.
Test statistic
SE = σ/√n = 5/√100 = 0.5
z = (499.1 − 500)/0.5 = −1.8
Tail area
Φ(1.8) = 0.9641
1 − Φ(1.8) = 0.0359
p-value
p = 2·0.0359 = 0.07186
Answer
p = 0.07186
p > 0.05, so H₀ is not rejected at the 5% level.
10.Hypothesis tests & confidence intervals
Answer: z = −2.667
Hypothesis test: the test statistic = (estimate − value under H₀)/(standard error under H₀). The p-value is the probability, assuming H₀ is true, of a statistic at least as extreme as the one observed: the area in one tail for a one-sided H₁, twice the tail area for a two-sided H₁. Reject H₀ when p ≤ α.
Sample proportions
p̂₁ = 165/300 = 0.55
p̂₂ = 83/120 = 0.6917
p̂ = 248/420 = 0.5905
Standard error under H₀
SE = √(0.5905·0.4095·(1/300 + 1/120)) = 0.05311
Test statistic
z = (0.55 − 0.6917)/0.05311 = −2.667
Answer
z = −2.667
|z| > 1.96, so H₀ is rejected at the 5% level (two-sided).
11.Linear regression & correlation
Answer: r = 0.9939
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 = 30/5 = 6
ȳ = 75/5 = 15
Sums of squares and products
Sxx = Σ(x − x̄)² = 40
Sxy = Σ(x − x̄)(y − ȳ) = 70
Syy = Σ(y − ȳ)² = 124
Correlation coefficient
r = 70/√(40·124) = 0.9939
Answer
r = 0.9939
12.Linear regression & correlation
Answer: e = 2.15
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 = 105/5 = 21
ȳ = 316/5 = 63.2
Sums of squares and products
Sxx = Σ(x − x̄)² = 90
Sxy = Σ(x − x̄)(y − ȳ) = 279
Regression line
b = 279/90 = 3.1
a = 63.2 − 3.1·21 = −1.9
ŷ = −1.9 + 3.1·x
Predicted value
ŷ = −1.9 + 3.1·22.5 = 67.85
Residual
e = 70 − 67.85 = 2.15
Answer
e = 2.15
The point lies above the line.
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