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, 56, 50, 53, 40.
n = 5
s =
2.Descriptive statisticsHard
A quantity is normally distributed with mean 500 and standard deviation 100. What proportion P of values is below 380?
μ = 500σ = 100x = 380
P =
3.ProbabilityMedium
In how many ways N can a committee of 3 people be chosen from a group of 15? The order does not matter.
n = 15k = 3
N =
4.ProbabilityHard
X has the distribution shown, and Y = 10X + 20. Find the standard deviation σY of Y.
x: 0, 2, 4, 6p(x): 0.3, 0.35, 0.25, 0.1Y = 10X + 20
σY =
5.Normal distributionMedium
The masses of bolts are independent and normal with mean 40 g and standard deviation 1.2 g. Find the probability P that the total mass of 16 of them exceeds 644.8 g. Round z to two decimals.
μ = 40 gσ = 1.2 gn = 16> 644.8 g
P =
6.Normal distributionHard
The bag weights are normally distributed with mean 250 g and standard deviation 5 g. Find the value x below which 25% of them lie (the 25th percentile).
μ = 250σ = 5P = 25%
x = g
7.Confidence intervalsMedium
A sample of 8 items has mean 119.5 and standard deviation s = 4.5. The population is roughly normal. Find the margin of error E of a 99% confidence interval for the mean.
n = 8x̄ = 119.5s = 4.599%t(7) = 3.499
E =
8.Confidence intervalsHard
The population standard deviation is 24. How large a sample n do you need so that a 99% confidence interval for the mean has a margin of error of at most 0.5?
σ = 24E = 0.599%
n =
9.Hypothesis tests & confidence intervalsMedium
Two teaching methods are compared. Group 1: 16 students, mean score 52.5, standard deviation 2. Group 2: 6 students, mean score 43.1, standard deviation 11. Assume equal population variances and use the pooled two-sample t test. Find the test statistic t for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (group 1 minus group 2).
n₁ = 16x̄₁ = 52.5s₁ = 2n₂ = 6x̄₂ = 43.1s₂ = 11
t =
10.Hypothesis tests & confidence intervalsHard
How large must a sample be so that the 95% confidence interval for a mean has a margin of error of at most 0.5? The standard deviation is 20. Find the smallest sample size n.
σ = 20E = 0.595%
n =
11.Linear regression & correlationMedium
The table gives 5 paired observations of x = temperature in °C and y = ice creams sold. Their sums are given. Find Pearson's correlation coefficient r from these sums.
n = 5Σx = 110Σy = 395Σx² = 2510Σy² = 32619Σxy = 9038
r =
12.Linear regression & correlationHard
The table gives 6 paired observations of x = hours studied and y = exam score. Use the least-squares line ŷ = a + b·x. A new observation (11, 107) is made. Find its residual e = y − ŷ.
n = 6(11, 107)
e =
3Solutions
1.Descriptive statistics
Answer: s = 18.77
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 = 209/5 = 41.8
Squared deviations
Σ(x − x̄)² = (−31.8)² + 14.2² + 8.2² + 11.2² + (−1.8)² = 1409
Standard deviation
s = √(1409/4) = 18.77
Answer
s = 18.77
2.Descriptive statistics
Answer: P = 0.1151
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 = −1.2
Standard normal table
Φ(−1.2) = 0.1151
P = 0.1151
Answer
P = 0.1151
3.Probability
Answer: N = 455
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.
Combinations
C(15, 3) = (15·14·13)/(3·2·1) = 455
Answer
N = 455
4.Probability
Answer: σY = 19.26
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.
Spread of X
μ = 2.3, Var(X) = 3.71, σX = 1.926
Linear change
σY = |10|·1.926 = 19.26
Answer
σY = 19.26
The mean does move: E(Y) = 10·2.3 + 20 = 43.
5.Normal distribution
Answer: P = 0.1587
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 = 640 g, σT = 1.2·√16 = 4.8 g
Standardise
z = (644.8 − 640)/4.8 = 1
Probability
P = 1 − Φ(1) = 0.1587
Answer
P = 0.1587
6.Normal distribution
Answer: x = 246.6 g
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) = 0.25 → z = −0.6745
Back to x
x = 250 + −0.6745·5 = 246.63 g
Answer
x = 246.63 g
7.Confidence intervals
Answer: E = 5.567
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 = 7, t = 3.499
Margin of error
E = 3.499·4.5/√8 = 5.567
Answer
E = 5.567
8.Confidence intervals
Answer: n = 15289
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 = (2.576·24/0.5)² = 15289
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 15289
Answer
n = 15289
9.Hypothesis tests & confidence intervals
Answer: t = 3.405
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.
Pooled variance
sp² = (15·2² + 5·11²)/20 = 33.25
sp = 5.766
Standard error
SE = 5.766·√(1/16 + 1/6) = 2.76
Test statistic
t = (52.5 − 43.1)/2.76 = 3.405
Answer
t = 3.405
With df = 20, the two-sided 5% critical value is 2.086; |t| is larger, so H₀ is rejected.
10.Hypothesis tests & confidence intervals
Answer: n = 6147
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).
Solve for n
n = (1.96·20/0.5)² = 6147
Round up
n = 6147
Answer
n = 6147
11.Linear regression & correlation
Answer: r = 0.9755
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.
Centred sums
Sxy = 9038 − 110·395/5 = 348
Sxx = 2510 − 110²/5 = 90
Syy = 32619 − 395²/5 = 1414
Correlation coefficient
r = 348/√(90·1414) = 0.9755
Answer
r = 0.9755
12.Linear regression & correlation
Answer: e = −5.49
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 = 42/6 = 7
ȳ = 545/6 = 90.83
Sums of squares and products
Sxx = Σ(x − x̄)² = 70
Sxy = Σ(x − x̄)(y − ȳ) = 379
Regression line
b = 379/70 = 5.414
a = 90.83 − 5.414·7 = 52.93
ŷ = 52.93 + 5.414·x
Predicted value
ŷ = 52.93 + 5.414·11 = 112.5
Residual
e = 107 − 112.5 = −5.49
Answer
e = −5.49
The point lies below the line.
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