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, 46, 51, 38, 18.
n = 5
s =
2.Descriptive statisticsHard
A quantity is normally distributed with mean 60 and standard deviation 8. What proportion P of values is below 68?
μ = 60σ = 8x = 68
P =
3.ProbabilityMedium
A bag holds 6 red and 6 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?
6 / 6
P =
4.ProbabilityHard
6 components are tested, and each one fails with probability 0.5, independently. What is the probability P that at most one fails?
n = 6p = 0.5
P =
5.Normal distributionMedium
The delivery times are normally distributed with mean 32 min and standard deviation 6 min. What fraction lies between 32.3 and 35 min? Give P as a decimal.
μ = 32σ = 6x1 = 32.3x2 = 35
P =
6.Normal distributionHard
The heights of adult men are normally distributed with mean 178 cm and standard deviation 7 cm. Find the value x below which 95% of them lie (the 95th percentile).
μ = 178σ = 7P = 95%
x = cm
7.Confidence intervalsMedium
A sample of 8 items has mean 55.5 and standard deviation s = 9.5. The population is roughly normal. Find the margin of error E of a 99% confidence interval for the mean.
n = 8x̄ = 55.5s = 9.599%t(7) = 3.499
E =
8.Confidence intervalsHard
The population standard deviation is 14. 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 = 14E = 2.590%
n =
9.Hypothesis tests & confidence intervalsMedium
15 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 3.41 and the standard deviation of the differences is 6.6. Find the paired test statistic t for H₀: μd = 0.
n = 15d̄ = 3.41sd = 6.6
t =
10.Hypothesis tests & confidence intervalsHard
A sample of 64 measurements has mean 73.1. The population standard deviation is 10. Find the lower limit L of the 90% confidence interval for the mean.
x̄ = 73.1σ = 10n = 6490%
L =
11.Linear regression & correlationMedium
The table gives 5 paired observations of x = car age in years and y = price in k€. Find Pearson's correlation coefficient r.
n = 5
r =
12.Linear regression & correlationHard
The table gives 6 paired observations of x = car age in years and y = price in k€. Use the least-squares line ŷ = a + b·x. For which value x does the line predict ŷ = 22?
n = 6ŷ = 22
x =
3Solutions
1.Descriptive statistics
Answer: s = 13.05
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 = 199/5 = 39.8
Squared deviations
Σ(x − x̄)² = 6.2² + 6.2² + 11.2² + (−1.8)² + (−21.8)² = 680.8
Standard deviation
s = √(680.8/4) = 13.05
Answer
s = 13.05
2.Descriptive statistics
Answer: P = 0.8413
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) = 0.8413
P = 0.8413
Answer
P = 0.8413
3.Probability
Answer: P = 0.2273
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 = 6/12 · 5/11 = 30/132 = 0.2273
Answer
P = 0.2273
4.Probability
Answer: P = 0.1094
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.
Zero and one failure
P(X = 0) = 0.56 = 0.01563
P(X = 1) = 6·0.5·0.55 = 0.09375
Add them
P = 0.01563 + 0.09375 = 0.1094
Answer
P = 0.1094
5.Normal distribution
Answer: P = 0.1715
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-scores
z₁ = (32.3 − 32)/6 = 0.05
z₂ = (35 − 32)/6 = 0.5
Area between
P = Φ(0.5) − Φ(0.05) = 0.6915 − 0.5199 = 0.1715
Answer
P = 0.1715
6.Normal distribution
Answer: x = 189.5 cm
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) = 0.95 → z = 1.645
Back to x
x = 178 + 1.645·7 = 189.51 cm
Answer
x = 189.51 cm
7.Confidence intervals
Answer: E = 11.75
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 = 3.499·9.5/√8 = 11.75
Answer
E = 11.75
8.Confidence intervals
Answer: n = 85
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 = (1.645·14/2.5)² = 84.86
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 85
Answer
n = 85
9.Hypothesis tests & confidence intervals
Answer: t = 2.001
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 = 6.6/√15 = 1.704
Test statistic
t = 3.41/1.704 = 2.001
Answer
t = 2.001
With df = 14, the two-sided 5% critical value is 2.145; |t| is smaller, so H₀ is not rejected.
10.Hypothesis tests & confidence intervals
Answer: L = 71.04
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 = 1.645·10/√64 = 2.056
Interval limit
L = 73.1 − 2.056 = 71.04
Answer
L = 71.04
11.Linear regression & correlation
Answer: r = −0.9138
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 = 94/5 = 18.8
Sums of squares and products
Sxx = Σ(x − x̄)² = 10
Sxy = Σ(x − x̄)(y − ȳ) = −18
Syy = Σ(y − ȳ)² = 38.8
Correlation coefficient
r = −18/√(10·38.8) = −0.9138
Answer
r = −0.9138
12.Linear regression & correlation
Answer: x = 1.361
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 = 21/6 = 3.5
ȳ = 110/6 = 18.33
Sums of squares and products
Sxx = Σ(x − x̄)² = 17.5
Sxy = Σ(x − x̄)(y − ȳ) = −30
Regression line
b = −30/17.5 = −1.714
a = 18.33 − (−1.714)·3.5 = 24.33
ŷ = 24.33 − 1.714·x
Solve for x
x = (22 − 24.33)/(−1.714) = 1.361
Answer
x = 1.361
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