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: 45, 47, 46, 51, 60.
n = 5
s =
2.Descriptive statisticsHard
Exam scores are normally distributed with mean 100 and standard deviation 5. Find the z-score z of a score of 103.
μ = 100σ = 5x = 103
z =
3.ProbabilityMedium
A bag holds 3 red and 3 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?
3 / 3
P =
4.ProbabilityHard
8 components are tested, and each one fails with probability 0.5, independently. What is the probability P that at most one fails?
n = 8p = 0.5
P =
5.Normal distributionMedium
Drag z so that the shaded area between −z and z under the standard normal curve is 0.9127. Find z to two decimals.
A = 0.9127
z =
6.Normal distributionHard
Hole diameters D are normal with mean 25.03 mm and standard deviation 0.01 mm. Shaft diameters d are independent and normal with mean 25 mm and standard deviation 0.015 mm. A randomly chosen shaft fits when D − d > 0. Find the probability P that it fits. Round z to two decimals.
hole: μ = 25.03 mm, σ = 0.01 mmshaft: μ = 25 mm, σ = 0.015 mm
P =
7.Confidence intervalsMedium
A sample of 6 items has mean 29 and standard deviation s = 6. The population is roughly normal. Find the upper limit of the 90% confidence interval for the mean.
n = 6x̄ = 29s = 690%t(5) = 2.015
upper limit =
8.Confidence intervalsHard
The population standard deviation is 21. How large a sample n do you need so that a 99% confidence interval for the mean has a margin of error of at most 1?
σ = 21E = 199%
n =
9.Hypothesis tests & confidence intervalsMedium
6 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 9.52 and the standard deviation of the differences is 10.6. Find the paired test statistic t for H₀: μd = 0.
n = 6d̄ = 9.52sd = 10.6
t =
10.Hypothesis tests & confidence intervalsHard
A sample of 36 measurements has mean 70.2. The population standard deviation is 10. Find the lower limit L of the 90% confidence interval for the mean.
x̄ = 70.2σ = 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 Pearson's correlation coefficient r.
n = 5
r =
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. A new observation (4, 60) is made. Find its residual e = y − ŷ.
n = 6(4, 60)
e =
3Solutions
1.Descriptive statistics
Answer: s = 6.14
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 = 249/5 = 49.8
Squared deviations
Σ(x − x̄)² = (−4.8)² + (−2.8)² + (−3.8)² + 1.2² + 10.2² = 150.8
Standard deviation
s = √(150.8/4) = 6.14
Answer
s = 6.14
2.Descriptive statistics
Answer: z = 0.6
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 = (103 − 100)/5 = 0.6
Answer
z = 0.6
3.Probability
Answer: P = 0.2
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 = 3/6 · 2/5 = 6/30 = 0.2
Answer
P = 0.2
4.Probability
Answer: P = 0.03516
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) = 0.58 = 0.003906
P(X = 1) = 8·0.5·0.57 = 0.03125
Add them
P = 0.003906 + 0.03125 = 0.03516
Answer
P = 0.03516
5.Normal distribution
Answer: z = 1.71
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 + 0.9127)/2 = 0.9564
Read the z-table backwards
Find 0.9564 inside the table and read off z.
z = 1.71
Answer
z = 1.71
6.Normal distribution
Answer: P = 0.9515
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 = 25.03 − 25 = 0.03 mm, σ = 0.01803 mm
Standardise
z = (0 − 0.03)/0.01803 = −1.66
Probability
P = 1 − Φ(−1.66) = 0.9515
Answer
P = 0.9515
7.Confidence intervals
Answer: upper limit = 33.94
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 = 5, t = 2.015
Margin of error
E = 2.015·6/√6 = 4.936
Interval
29 ± 4.936 → [24.06, 33.94]
Answer
upper limit = 33.94
8.Confidence intervals
Answer: n = 2927
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 = (2.576·21/1)² = 2926
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 2927
Answer
n = 2927
9.Hypothesis tests & confidence intervals
Answer: t = 2.2
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 = 10.6/√6 = 4.327
Test statistic
t = 9.52/4.327 = 2.2
Answer
t = 2.2
With df = 5, the two-sided 5% critical value is 2.571; |t| is smaller, so H₀ is not rejected.
10.Hypothesis tests & confidence intervals
Answer: L = 67.46
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 = 70.2 − 2.742 = 67.46
Answer
L = 67.46
11.Linear regression & correlation
Answer: r = 0.9953
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 = 25/5 = 5
ȳ = 231/5 = 46.2
Sums of squares and products
Sxx = Σ(x − x̄)² = 40
Sxy = Σ(x − x̄)(y − ȳ) = 136
Syy = Σ(y − ȳ)² = 466.8
Correlation coefficient
r = 136/√(40·466.8) = 0.9953
Answer
r = 0.9953
12.Linear regression & correlation
Answer: e = 2.095
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 = 48/6 = 8
ȳ = 464/6 = 77.33
Sums of squares and products
Sxx = Σ(x − x̄)² = 70
Sxy = Σ(x − x̄)(y − ȳ) = 340
Regression line
b = 340/70 = 4.857
a = 77.33 − 4.857·8 = 38.48
ŷ = 38.48 + 4.857·x
Predicted value
ŷ = 38.48 + 4.857·4 = 57.9
Residual
e = 60 − 57.9 = 2.095
Answer
e = 2.095
The point lies above the line.
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