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: 59, 40, 41, 42, 42, 15, 41.
n = 7
s =
2.Descriptive statisticsHard
Exam scores are normally distributed with mean 500 and standard deviation 100. Find the z-score z of a score of 625.
μ = 500σ = 100x = 625
z =
3.ProbabilityMedium
A bag holds 3 red and 5 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?
3 / 5
P =
4.ProbabilityHard
A test has 5 multiple-choice questions. A student who guesses gets each one right with probability 0.6, independently. What is the probability P of exactly 3 correct answers?
n = 5p = 0.6k = 3
P =
5.Normal distributionMedium
X is normal with mean 120 mm and standard deviation 0.3 mm; Y is independent and normal with mean 80 mm and standard deviation 0.4 mm. Find the probability P that X + Y is more than 199.3 mm. Round z to two decimals.
μX = 120, σX = 0.3μY = 80, σY = 0.4X + Y > 199.3 mm
P =
6.Normal distributionHard
The exam scores are normally distributed with mean 65 points and standard deviation 10 points. Find the value x below which 90% of them lie (the 90th percentile).
μ = 65σ = 10P = 90%
x = points
7.Confidence intervalsMedium
A sample of 9 items has mean 99.5 and standard deviation s = 12.5. The population is roughly normal. Find the upper limit of the 90% confidence interval for the mean.
n = 9x̄ = 99.5s = 12.590%t(8) = 1.86
upper limit =
8.Confidence intervalsHard
The population standard deviation is 27. 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 = 27E = 2.590%
n =
9.Hypothesis tests & confidence intervalsMedium
10 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 4.18 and the standard deviation of the differences is 4.9. Find the paired test statistic t for H₀: μd = 0.
n = 10d̄ = 4.18sd = 4.9
t =
10.Hypothesis tests & confidence intervalsHard
A one-sided z test of H₀: μ = 500 against H₁: μ > 500 is done at α = 0.01 with a sample of 64; the standard deviation is known to be 24. In reality the mean is 504.5. Find the power 1 − β of the test in percent.
μ₀ = 500μ = 504.5σ = 24n = 64α = 0.01
1 − β = %
11.Linear regression & correlationMedium
The table gives 6 paired observations of x = hours studied and y = exam score. Find the coefficient of determination r² of the least-squares line.
n = 6
r² =
12.Linear regression & correlationHard
The table gives 5 paired observations of x = car age in years and y = price in k€. Use the least-squares line ŷ = a + b·x. A new observation (1.5, 22) is made. Find its residual e = y − ŷ.
n = 5(1.5, 22)
e =
3Solutions
1.Descriptive statistics
Answer: s = 12.88
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 = 280/7 = 40
Squared deviations
Σ(x − x̄)² = 19² + 0² + 1² + 2² + 2² + (−25)² + 1² = 996
Standard deviation
s = √(996/6) = 12.88
Answer
s = 12.88
2.Descriptive statistics
Answer: z = 1.25
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 = (625 − 500)/100 = 1.25
Answer
z = 1.25
3.Probability
Answer: P = 0.1071
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 = 3/8 · 2/7 = 6/56 = 0.1071
Answer
P = 0.1071
4.Probability
Answer: P = 0.3456
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.
Binomial formula
P(X = 3) = 10·0.63·0.42 = 10·0.216·0.16 = 0.3456
Answer
P = 0.3456
5.Normal distribution
Answer: P = 0.9192
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 sum
μ = 200 mm, σ = √(0.3² + 0.4²) = 0.5 mm
Standardise
z = (199.3 − 200)/0.5 = −1.4
Upper tail
P = 1 − Φ(−1.4) = 1 − 0.08076 = 0.9192
Answer
P = 0.9192
6.Normal distribution
Answer: x = 77.82 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 = 65 + 1.282·10 = 77.82 points
Answer
x = 77.82 points
7.Confidence intervals
Answer: upper limit = 107.3
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 = 8, t = 1.86
Margin of error
E = 1.86·12.5/√9 = 7.75
Interval
99.5 ± 7.75 → [91.75, 107.3]
Answer
upper limit = 107.3
8.Confidence intervals
Answer: n = 316
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 = (1.645·27/2.5)² = 315.6
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 316
Answer
n = 316
9.Hypothesis tests & confidence intervals
Answer: t = 2.698
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 = 4.9/√10 = 1.55
Test statistic
t = 4.18/1.55 = 2.698
Answer
t = 2.698
With df = 9, the two-sided 5% critical value is 2.262; |t| is larger, so H₀ is rejected.
10.Hypothesis tests & confidence intervals
Answer: 1 − β = 20.44 %
Power of a test: power = 1 − β is the probability of rejecting H₀ when it is false. Find the cutoff x̄c from μ₀ and z_α, then compute the probability of landing beyond x̄c using the true mean: z = (x̄c − μ)/(σ/√n).
Critical sample mean
SE = 24/√64 = 3
x̄c = 500 + 2.326·3 = 507
z under the true mean
z = (507 − 504.5)/3 = 0.826
Power
power = P(Z > 0.826) = 20.44%
1 − β = 20.44%
Answer
1 − β = 20.44%
11.Linear regression & correlation
Answer: r² = 0.7638
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 = 337/6 = 56.17
Sums of squares and products
Sxx = Σ(x − x̄)² = 17.5
Sxy = Σ(x − x̄)(y − ȳ) = 56.5
Syy = Σ(y − ȳ)² = 238.8
Coefficient of determination
r² = 56.5²/(17.5·238.8) = 0.7638
Answer
r² = 0.7638
About 76.38% of the variation in y is explained by the line.
12.Linear regression & correlation
Answer: e = −3.8
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 = 117/5 = 23.4
Sums of squares and products
Sxx = Σ(x − x̄)² = 10
Sxy = Σ(x − x̄)(y − ȳ) = −16
Regression line
b = −16/10 = −1.6
a = 23.4 − (−1.6)·3 = 28.2
ŷ = 28.2 − 1.6·x
Predicted value
ŷ = 28.2 + (−1.6)·1.5 = 25.8
Residual
e = 22 − 25.8 = −3.8
Answer
e = −3.8
The point lies below the line.
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