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: 27, 16, 16, 50, 19, 35, 15.
n = 7
s =
2.Descriptive statisticsHard
A quantity is normally distributed with mean 50 and standard deviation 10. What proportion P of values is below 56?
μ = 50σ = 10x = 56
P =
3.ProbabilityMedium
A bag holds 6 red and 8 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?
6 / 8
P =
4.ProbabilityHard
12 components are tested, and each one fails with probability 0.7, independently. What is the probability P that at most one fails?
n = 12p = 0.7
P =
5.Normal distributionMedium
The battery lifetimes are normally distributed with mean 500 h and standard deviation 40 h. What fraction lies between 436 and 580 h? Give P as a decimal.
μ = 500σ = 40x1 = 436x2 = 580
P =
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 0.02 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 = 0.02 mm
P =
7.Confidence intervalsMedium
A sample of 13 items has mean 48 and standard deviation s = 10.5. The population is roughly normal. Find the lower limit of the 95% confidence interval for the mean.
n = 13x̄ = 48s = 10.595%t(12) = 2.179
lower limit =
8.Confidence intervalsHard
The population standard deviation is 15. 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 = 15E = 2.590%
n =
9.Hypothesis tests & confidence intervalsMedium
A sample of 36 students has a mean score of 494.8 and a sample standard deviation of 12. Find the test statistic t for H₀: μ = 500.
μ₀ = 500x̄ = 494.8s = 12n = 36
t =
10.Hypothesis tests & confidence intervalsHard
A sample of 36 measurements has mean 124. The population standard deviation is 10. Find the lower limit L of the 90% confidence interval for the mean.
x̄ = 124σ = 10n = 3690%
L =
11.Linear regression & correlationMedium
The table gives 5 paired observations of x = advertising spend in k€ and y = sales in k€. Find the coefficient of determination r² of the least-squares line.
n = 5
r² =
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. Predict ŷ for x = 19.
n = 5x = 19
ŷ =
3Solutions
1.Descriptive statistics
Answer: s = 13.07
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 = 178/7 = 25.43
Squared deviations
Σ(x − x̄)² = 1.571² + (−9.429)² + (−9.429)² + 24.57² + (−6.429)² + 9.571² + (−10.43)² = 1026
Standard deviation
s = √(1026/6) = 13.07
Answer
s = 13.07
2.Descriptive statistics
Answer: P = 0.7257
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 = 0.6
Standard normal table
Φ(0.6) = 0.7257
P = 0.7257
Answer
P = 0.7257
3.Probability
Answer: P = 0.1648
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 = 6/14 · 5/13 = 30/182 = 0.1648
Answer
P = 0.1648
4.Probability
Answer: P = 1.541 × 10⁻⁵
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) = 0.312 = 5.314 × 10⁻⁷
P(X = 1) = 12·0.7·0.311 = 1.488 × 10⁻⁵
Add them
P = 5.314 × 10⁻⁷ + 1.488 × 10⁻⁵ = 1.541 × 10⁻⁵
Answer
P = 1.541 × 10⁻⁵
5.Normal distribution
Answer: P = 0.9225
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 = (436 − 500)/40 = −1.6
z₂ = (580 − 500)/40 = 2
Area between
P = Φ(2) − Φ(−1.6) = 0.9772 − 0.0548 = 0.9225
Answer
P = 0.9225
6.Normal distribution
Answer: P = 0.9772
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 = 0.025 mm
Standardise
z = (0 − 0.05)/0.025 = −2
Probability
P = 1 − Φ(−2) = 0.9772
Answer
P = 0.9772
7.Confidence intervals
Answer: lower limit = 41.65
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 = 12, t = 2.179
Margin of error
E = 2.179·10.5/√13 = 6.346
Interval
48 ± 6.346 → [41.65, 54.35]
Answer
lower limit = 41.65
8.Confidence intervals
Answer: n = 98
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 = (1.645·15/2.5)² = 97.42
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 98
Answer
n = 98
9.Hypothesis tests & confidence intervals
Answer: t = −2.6
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
SE = 12/√36 = 12/6 = 2
Test statistic
t = (494.8 − 500)/2 = −2.6
Answer
t = −2.6
t0.025; df = 35 = 2.03
|t| > 2.03, so H₀ is rejected at the 5% level (two-sided).
10.Hypothesis tests & confidence intervals
Answer: L = 121.3
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/√36 = 2.742
Interval limit
L = 124 − 2.742 = 121.3
Answer
L = 121.3
11.Linear regression & correlation
Answer: r² = 0.9412
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 = 20/5 = 4
ȳ = 233/5 = 46.6
Sums of squares and products
Sxx = Σ(x − x̄)² = 10
Sxy = Σ(x − x̄)(y − ȳ) = 56
Syy = Σ(y − ȳ)² = 333.2
Coefficient of determination
r² = 56²/(10·333.2) = 0.9412
Answer
r² = 0.9412
About 94.12% of the variation in y is explained by the line.
12.Linear regression & correlation
Answer: ŷ = 76.35
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 = 100/5 = 20
ȳ = 407/5 = 81.4
Sums of squares and products
Sxx = Σ(x − x̄)² = 40
Sxy = Σ(x − x̄)(y − ȳ) = 202
Regression line
b = 202/40 = 5.05
a = 81.4 − 5.05·20 = −19.6
ŷ = −19.6 + 5.05·x
Prediction
ŷ = −19.6 + 5.05·19 = 76.35
Answer
ŷ = 76.35
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