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: 36, 19, 59, 10, 13, 36.
n = 6
s =
2.Descriptive statisticsHard
Exam scores are normally distributed with mean 70 and standard deviation 15. Find the z-score z of a score of 62.5.
μ = 70σ = 15x = 62.5
z =
3.ProbabilityMedium
In how many ways N can a committee of 5 people be chosen from a group of 12? The order does not matter.
n = 12k = 5
N =
4.ProbabilityHard
9 components are tested, and each one fails with probability 0.4, independently. What is the probability P that at most one fails?
n = 9p = 0.4
P =
5.Normal distributionMedium
The IQ scores are normally distributed with mean 100 points and standard deviation 15 points. What fraction lies between 88.75 and 112 points? Give P as a decimal.
μ = 100σ = 15x1 = 88.75x2 = 112
P =
6.Normal distributionHard
The IQ scores are normally distributed with mean 100 points and standard deviation 15 points. Find the value x below which 90% of them lie (the 90th percentile).
μ = 100σ = 15P = 90%
x = points
7.Confidence intervalsMedium
The battery life of two phone models is compared. Model 1: 60 phones, mean 107 h, known standard deviation 13 h. Model 2: 100 phones, mean 103.5 h, known standard deviation 20 h. Find the margin of error E of the 90% confidence interval for μ₁ − μ₂.
n₁ = 60x̄₁ = 107 hσ₁ = 13 hn₂ = 100x̄₂ = 103.5 hσ₂ = 20 h90%
E = h
8.Confidence intervalsHard
The population standard deviation is 39. 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 5?
σ = 39E = 595%
n =
9.Hypothesis tests & confidence intervalsMedium
Two production lines make the same part. Line 1: a sample of 80 parts has mean length 129 mm, and the line's known standard deviation is 3 mm. Line 2: a sample of 50 parts has mean 125.1 mm, known standard deviation 15 mm. Find the test statistic z for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (line 1 minus line 2).
n₁ = 80x̄₁ = 129 mmσ₁ = 3 mmn₂ = 50x̄₂ = 125.1 mmσ₂ = 15 mm
z =
10.Hypothesis tests & confidence intervalsHard
A sample of 64 measurements has mean 198.7. The population standard deviation is 4. Find the lower limit L of the 99% confidence interval for the mean.
x̄ = 198.7σ = 4n = 6499%
L =
11.Linear regression & correlationMedium
The table gives 5 paired observations of x = temperature in °C and y = ice creams sold. 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 = load in kN and y = extension in mm. Use the least-squares line ŷ = a + b·x. A new observation (5, 17) is made. Find its residual e = y − ŷ.
n = 5(5, 17)
e =
3Solutions
1.Descriptive statistics
Answer: s = 18.52
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 = 173/6 = 28.83
Squared deviations
Σ(x − x̄)² = 7.167² + (−9.833)² + 30.17² + (−18.83)² + (−15.83)² + 7.167² = 1715
Standard deviation
s = √(1715/5) = 18.52
Answer
s = 18.52
2.Descriptive statistics
Answer: z = −0.5
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 = (62.5 − 70)/15 = −0.5
Answer
z = −0.5
3.Probability
Answer: N = 792
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(12, 5) = (12·11·10·9·8)/(5·4·3·2·1) = 792
Answer
N = 792
4.Probability
Answer: P = 0.07054
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) = 0.69 = 0.01008
P(X = 1) = 9·0.4·0.68 = 0.06047
Add them
P = 0.01008 + 0.06047 = 0.07054
Answer
P = 0.07054
5.Normal distribution
Answer: P = 0.5615
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 = (88.75 − 100)/15 = −0.75
z₂ = (112 − 100)/15 = 0.8
Area between
P = Φ(0.8) − Φ(−0.75) = 0.7881 − 0.2266 = 0.5615
Answer
P = 0.5615
6.Normal distribution
Answer: x = 119.2 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) = 0.9 → z = 1.282
Back to x
x = 100 + 1.282·15 = 119.22 points
Answer
x = 119.22 points
7.Confidence intervals
Answer: E = 4.295 h
Interval for a difference: estimate ± critical value × standard error. For two independent means with known σ: SE = √(σ₁²/n₁ + σ₂²/n₂) and z. With equal unknown variances: pooled s_p and t with n₁ + n₂ − 2 degrees of freedom. For paired data, work with the differences: d̄ ± t·s_d/√n.
Point estimate
x̄₁ − x̄₂ = 107 − 103.5 = 3.5
Standard error
SE = √(13²/60 + 20²/100) = 2.611
Margin of error
E = 1.645·2.611 = 4.295
Answer
E = 4.295 h
The interval contains 0, so the data do not show a significant difference.
8.Confidence intervals
Answer: n = 234
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 = (1.96·39/5)² = 233.7
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 234
Answer
n = 234
9.Hypothesis tests & confidence intervals
Answer: z = 1.816
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.
Standard error of the difference
SE = √(3²/80 + 15²/50) = √4.613 = 2.148
Test statistic
z = (129 − 125.1)/2.148 = 1.816
Answer
z = 1.816
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).
10.Hypothesis tests & confidence intervals
Answer: L = 197.4
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 = 2.576·4/√64 = 1.288
Interval limit
L = 198.7 − 1.288 = 197.4
Answer
L = 197.4
11.Linear regression & correlation
Answer: r² = 0.9309
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 = 331/5 = 66.2
Sums of squares and products
Sxx = Σ(x − x̄)² = 40
Sxy = Σ(x − x̄)(y − ȳ) = 198
Syy = Σ(y − ȳ)² = 1053
Coefficient of determination
r² = 198²/(40·1053) = 0.9309
Answer
r² = 0.9309
About 93.09% of the variation in y is explained by the line.
12.Linear regression & correlation
Answer: e = −2.3
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 = 116/5 = 23.2
Sums of squares and products
Sxx = Σ(x − x̄)² = 40
Sxy = Σ(x − x̄)(y − ȳ) = 156
Regression line
b = 156/40 = 3.9
a = 23.2 − 3.9·6 = −0.2
ŷ = −0.2 + 3.9·x
Predicted value
ŷ = −0.2 + 3.9·5 = 19.3
Residual
e = 17 − 19.3 = −2.3
Answer
e = −2.3
The point lies below the line.
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