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: 47, 60, 42, 11, 12, 39.
n = 6
s =
2.Descriptive statisticsHard
A quantity is normally distributed with mean 170 and standard deviation 8. What proportion P of values is below 180?
μ = 170σ = 8x = 180
P =
3.ProbabilityMedium
For the distribution shown, find the variance Var(X) of X.
x: 0, 2, 4, 6p(x): 0.25, 0.35, 0.3, 0.1
Var(X) =
4.ProbabilityHard
X has the distribution shown, and Y = 10X − 3. Find the standard deviation σY of Y.
x: 10, 20, 30, 40p(x): 0.3, 0.05, 0.35, 0.3Y = 10X − 3
σY =
5.Normal distributionMedium
The bag weights are normally distributed with mean 250 g and standard deviation 5 g. What fraction lies between 250 and 252.75 g? Give P as a decimal.
μ = 250σ = 5x1 = 250x2 = 252.75
P =
6.Normal distributionHard
The IQ scores are normally distributed with mean 100 points and standard deviation 15 points. Drag the cut-off so the shaded top 95% is to the right of it. Find the cut-off x.
μ = 100 pointsσ = 15 points95%
x = points
7.Confidence intervalsMedium
A sample of 15 items has mean 29 and standard deviation s = 10.5. The population is roughly normal. Find the margin of error E of a 99% confidence interval for the mean.
n = 15x̄ = 29s = 10.599%t(14) = 2.977
E =
8.Confidence intervalsHard
The population standard deviation is 12. 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 3?
σ = 12E = 390%
n =
9.Hypothesis tests & confidence intervalsMedium
Two teaching methods are compared. Group 1: 13 students, mean score 88, standard deviation 4. Group 2: 9 students, mean score 77.7, standard deviation 9.5. 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 = 13x̄₁ = 88s₁ = 4n₂ = 9x̄₂ = 77.7s₂ = 9.5
t =
10.Hypothesis tests & confidence intervalsHard
How large must a sample be so that the 99% confidence interval for a mean has a margin of error of at most 1.5? The standard deviation is 10. Find the smallest sample size n.
σ = 10E = 1.599%
n =
11.Linear regression & correlationMedium
The table gives 6 paired observations of x = load in kN and y = extension in mm. 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 = temperature in °C and y = ice creams sold. Use the least-squares line ŷ = a + b·x. A new observation (26, 85) is made. Find its residual e = y − ŷ.
n = 6(26, 85)
e =
3Solutions
1.Descriptive statistics
Answer: s = 19.69
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 = 211/6 = 35.17
Squared deviations
Σ(x − x̄)² = 11.83² + 24.83² + 6.833² + (−24.17)² + (−23.17)² + 3.833² = 1939
Standard deviation
s = √(1939/5) = 19.69
Answer
s = 19.69
2.Descriptive statistics
Answer: P = 0.8944
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 = 1.25
Standard normal table
Φ(1.25) = 0.8944
P = 0.8944
Answer
P = 0.8944
3.Probability
Answer: Var(X) = 3.55
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.
Mean
μ = 0·0.25 + 2·0.35 + 4·0.3 + 6·0.1 = 2.5
Mean of the squares
E(X²) = 0·0.25 + 4·0.35 + 16·0.3 + 36·0.1 = 9.8
Variance
Var(X) = 9.8 − 2.5² = 3.55
Answer
Var(X) = 3.55
4.Probability
Answer: σY = 119.5
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 = 26.5, Var(X) = 142.8, σX = 11.95
Linear change
σY = |10|·11.95 = 119.5
Answer
σY = 119.5
The mean does move: E(Y) = 10·26.5 − 3 = 262.
5.Normal distribution
Answer: P = 0.2088
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 = (250 − 250)/5 = 0
z₂ = (252.75 − 250)/5 = 0.55
Area between
P = Φ(0.55) − Φ(0) = 0.7088 − 0.5 = 0.2088
Answer
P = 0.2088
6.Normal distribution
Answer: x = 75.4 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) = 0.05
z from the table
z ≈ −1.64
Back to the original units
x = 100 − 1.64·15 = 75.4 points
Answer
x = 75.4 points
7.Confidence intervals
Answer: E = 8.071
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 = 14, t = 2.977
Margin of error
E = 2.977·10.5/√15 = 8.071
Answer
E = 8.071
8.Confidence intervals
Answer: n = 44
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 = (1.645·12/3)² = 43.3
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 44
Answer
n = 44
9.Hypothesis tests & confidence intervals
Answer: t = 3.514
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 = (12·4² + 8·9.5²)/20 = 45.7
sp = 6.76
Standard error
SE = 6.76·√(1/13 + 1/9) = 2.931
Test statistic
t = (88 − 77.7)/2.931 = 3.514
Answer
t = 3.514
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 = 295
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 = (2.576·10/1.5)² = 294.9
Round up
n = 295
Answer
n = 295
11.Linear regression & correlation
Answer: r² = 0.9577
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 = 70/6 = 11.67
Sums of squares and products
Sxx = Σ(x − x̄)² = 70
Sxy = Σ(x − x̄)(y − ȳ) = 104
Syy = Σ(y − ȳ)² = 161.3
Coefficient of determination
r² = 104²/(70·161.3) = 0.9577
Answer
r² = 0.9577
About 95.77% of the variation in y is explained by the line.
12.Linear regression & correlation
Answer: e = 1.805
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 = 132/6 = 22
ȳ = 433/6 = 72.17
Sums of squares and products
Sxx = Σ(x − x̄)² = 70
Sxy = Σ(x − x̄)(y − ȳ) = 193
Regression line
b = 193/70 = 2.757
a = 72.17 − 2.757·22 = 11.51
ŷ = 11.51 + 2.757·x
Predicted value
ŷ = 11.51 + 2.757·26 = 83.2
Residual
e = 85 − 83.2 = 1.805
Answer
e = 1.805
The point lies above the line.
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