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: 46, 22, 45, 24, 45.
n = 5
s =
2.Descriptive statisticsHard
A quantity is normally distributed with mean 60 and standard deviation 10. What proportion P of values is below 50?
μ = 60σ = 10x = 50
P =
3.ProbabilityMedium
In how many ways N can a committee of 3 people be chosen from a group of 13? The order does not matter.
n = 13k = 3
N =
4.ProbabilityHard
A test has 5 multiple-choice questions. A student who guesses gets each one right with probability 0.7, independently. What is the probability P of exactly 1 correct answers?
n = 5p = 0.7k = 1
P =
5.Normal distributionMedium
Drag z so that the shaded area between −z and z under the standard normal curve is 0.2737. Find z to two decimals.
A = 0.2737
z =
6.Normal distributionHard
The exam scores are normally distributed with mean 65 points and standard deviation 10 points. Drag the cut-off so the shaded top 60% is to the right of it. Find the cut-off x.
μ = 65 pointsσ = 10 points60%
x = points
7.Confidence intervalsMedium
A sample of 8 items has mean 14 and standard deviation s = 19.5. The population is roughly normal. Find the upper limit of the 95% confidence interval for the mean.
n = 8x̄ = 14s = 19.595%t(7) = 2.365
upper limit =
8.Confidence intervalsHard
The population standard deviation is 24. 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?
σ = 24E = 290%
n =
9.Hypothesis tests & confidence intervalsMedium
The mean of a population is claimed to be 500; the population standard deviation is known to be 20. A random sample of 25 observations has a mean of 507.68. Test H₀: μ = 500 against H₁: μ ≠ 500. Find the p-value p.
μ₀ = 500σ = 20n = 25x̄ = 507.68H₁: μ ≠ μ₀
p =
10.Hypothesis tests & confidence intervalsHard
A sample of 25 measurements has mean 196.1. The population standard deviation is 15. Find the upper limit U of the 90% confidence interval for the mean.
x̄ = 196.1σ = 15n = 2590%
U =
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 6 paired observations of x = advertising spend in k€ and y = sales in k€. Use the least-squares line ŷ = a + b·x. Predict ŷ for x = 3.
n = 6x = 3
ŷ =
3Solutions
1.Descriptive statistics
Answer: s = 12.26
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 = 182/5 = 36.4
Squared deviations
Σ(x − x̄)² = 9.6² + (−14.4)² + 8.6² + (−12.4)² + 8.6² = 601.2
Standard deviation
s = √(601.2/4) = 12.26
Answer
s = 12.26
2.Descriptive statistics
Answer: P = 0.1587
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
Standard normal table
Φ(−1) = 0.1587
P = 0.1587
Answer
P = 0.1587
3.Probability
Answer: N = 286
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(13, 3) = (13·12·11)/(3·2·1) = 286
Answer
N = 286
4.Probability
Answer: P = 0.02835
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 = 1) = 5·0.71·0.34 = 5·0.7·0.0081 = 0.02835
Answer
P = 0.02835
5.Normal distribution
Answer: z = 0.35
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 + 0.2737)/2 = 0.6368
Read the z-table backwards
Find 0.6368 inside the table and read off z.
z = 0.35
Answer
z = 0.35
6.Normal distribution
Answer: x = 62.5 points
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
Φ(z) = 0.4
z from the table
z ≈ −0.25
Back to the original units
x = 65 − 0.25·10 = 62.5 points
Answer
x = 62.5 points
7.Confidence intervals
Answer: upper limit = 30.3
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 = 2.365
Margin of error
E = 2.365·19.5/√8 = 16.3
Interval
14 ± 16.3 → [−2.305, 30.3]
Answer
upper limit = 30.3
8.Confidence intervals
Answer: n = 390
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 = (1.645·24/2)² = 389.7
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 390
Answer
n = 390
9.Hypothesis tests & confidence intervals
Answer: p = 0.05486
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 = 20/√25 = 4
z = (507.68 − 500)/4 = 1.92
Tail area
Φ(1.92) = 0.9726
1 − Φ(1.92) = 0.0274
p-value
p = 2·0.0274 = 0.05486
Answer
p = 0.05486
p > 0.05, so H₀ is not rejected at the 5% level.
10.Hypothesis tests & confidence intervals
Answer: U = 201
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).
Margin of error
E = 1.645·15/√25 = 4.935
Interval limit
U = 196.1 + 4.935 = 201
Answer
U = 201
11.Linear regression & correlation
Answer: r² = 0.9659
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 = 147/6 = 24.5
ȳ = 480/6 = 80
Sums of squares and products
Sxx = Σ(x − x̄)² = 157.5
Sxy = Σ(x − x̄)(y − ȳ) = 609
Syy = Σ(y − ȳ)² = 2438
Coefficient of determination
r² = 609²/(157.5·2438) = 0.9659
Answer
r² = 0.9659
About 96.59% of the variation in y is explained by the line.
12.Linear regression & correlation
Answer: ŷ = 44.97
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 = 373/6 = 62.17
Sums of squares and products
Sxx = Σ(x − x̄)² = 70
Sxy = Σ(x − x̄)(y − ȳ) = 301
Regression line
b = 301/70 = 4.3
a = 62.17 − 4.3·7 = 32.07
ŷ = 32.07 + 4.3·x
Prediction
ŷ = 32.07 + 4.3·3 = 44.97
Answer
ŷ = 44.97
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