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
Research Methods and Statistics for Psychology: cram pack
Formula sheets, the key method steps for every topic and 16 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
Chi-square tests
- Test statistic
- χ² = Σ(O − E)²/E
- Goodness of fit
- E = n·p, df = k − 1 − (estimated parameters)
- Independence
- E = row total × column total / grand total, df = (r − 1)(c − 1)
- Critical values (α = 0.05)
- df 1: 3.841, 2: 5.991, 3: 7.815, 4: 9.488
- Decision
- reject H0 if χ² > χ²crit
2Practice problems
1.Research methods in psychologyMedium
Standardised instructions are used in an experiment so that:
- AThe results become statistically significant
- BThe researcher can adapt the procedure to suit each participant's needs
- CAll participants receive the same information in the same way
- DParticipants can choose their own conditions
2.Research methods in psychologyHard
Why do many researchers argue that p-values should be reported together with effect sizes and confidence intervals?
- AEffect sizes make sample size irrelevant to significance
- BP-values show the probability the hypothesis is true
- CA p-value does not show how large or precise the effect is
- DConfidence intervals are only needed in qualitative research
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 27, 23, 28, 24, 52, 14.
n = 6
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 100 and standard deviation 20. What proportion P of values is above 112?
μ = 100σ = 20x = 112
P =
5.ProbabilityMedium
A bag holds 3 red and 8 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?
3 / 8
P =
6.ProbabilityHard
X has the distribution shown, and Y = 5X + 10. Find the standard deviation σY of Y.
x: 1, 2, 3, 4, 5p(x): 0.25, 0.25, 0.2, 0.1, 0.2Y = 5X + 10
σY =
7.Normal distributionMedium
Drag z so that the shaded area under the standard normal curve, to the right of z, is 0.0375. Find z to two decimals.
A = 0.0375
z =
8.Normal distributionHard
The bag weights are normally distributed with mean 250 g and standard deviation 5 g. Find the value x below which 75% of them lie (the 75th percentile).
μ = 250σ = 5P = 75%
x = g
9.Confidence intervalsMedium
A sample of 12 items has mean 109.5 and standard deviation s = 7.5. The population is roughly normal. Find the upper limit of the 95% confidence interval for the mean.
n = 12x̄ = 109.5s = 7.595%t(11) = 2.201
upper limit =
10.Confidence intervalsHard
The population standard deviation is 31. 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 = 31E = 199%
n =
11.Hypothesis tests & confidence intervalsMedium
A sample of 49 students has a mean score of 69.86 and a sample standard deviation of 20. Find the test statistic t for H₀: μ = 75.
μ₀ = 75x̄ = 69.86s = 20n = 49
t =
12.Hypothesis tests & confidence intervalsHard
A sample of 81 measurements has mean 47.7. The population standard deviation is 8. Find the upper limit U of the 95% confidence interval for the mean.
x̄ = 47.7σ = 8n = 8195%
U =
13.Linear regression & correlationMedium
The table gives 6 paired observations of x = hours studied and y = exam score. Find Pearson's correlation coefficient r.
n = 6
r =
14.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. For which value x does the line predict ŷ = 59?
n = 6ŷ = 59
x =
15.Chi-square testsMedium
The contingency table shows observed counts. Compute the chi-square statistic χ² for the test of independence between the row and column variables. Do not use a continuity correction.
N = 2002 × 3
χ² =
16.Chi-square testsHard
A chi-square test of independence is carried out on a contingency table with 2 rows and 11 columns (not counting the totals). Using the table, find the critical value χ²crit at significance level α = 0.01.
α = 0.01
χ²crit =
3Solutions
1.Research methods in psychology
Answer: C · All participants receive the same information in the same way
Why this is the answer
Standardisation removes variation in procedure as an extraneous variable and makes the study replicable.
Answer
C · All participants receive the same information in the same way
2.Research methods in psychology
Answer: C · A p-value does not show how large or precise the effect is
Why this is the answer
A p-value depends heavily on sample size and only indicates how surprising the data are under the null. Effect sizes show magnitude and intervals show precision.
Answer
C · A p-value does not show how large or precise the effect is
3.Descriptive statistics
Answer: s = 12.76
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 = 168/6 = 28
Squared deviations
Σ(x − x̄)² = (−1)² + (−5)² + 0² + (−4)² + 24² + (−14)² = 814
Standard deviation
s = √(814/5) = 12.76
Answer
s = 12.76
4.Descriptive statistics
Answer: P = 0.2743
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 = 1 − 0.7257 = 0.2743
Answer
P = 0.2743
5.Probability
Answer: P = 0.05455
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 = 3/11 · 2/10 = 6/110 = 0.05455
Answer
P = 0.05455
6.Probability
Answer: σY = 7.224
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 = 2.75, Var(X) = 2.088, σX = 1.445
Linear change
σY = |5|·1.445 = 7.224
Answer
σY = 7.224
The mean does move: E(Y) = 5·2.75 + 10 = 23.75.
7.Normal distribution
Answer: z = 1.78
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 − 0.0375 = 0.9625
Read the z-table backwards
Find 0.9625 inside the table and read off z.
z = 1.78
Answer
z = 1.78
8.Normal distribution
Answer: x = 253.4 g
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) = 0.75 → z = 0.6745
Back to x
x = 250 + 0.6745·5 = 253.37 g
Answer
x = 253.37 g
9.Confidence intervals
Answer: upper limit = 114.3
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 = 11, t = 2.201
Margin of error
E = 2.201·7.5/√12 = 4.765
Interval
109.5 ± 4.765 → [104.7, 114.3]
Answer
upper limit = 114.3
10.Confidence intervals
Answer: n = 6377
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 = (2.576·31/1)² = 6377
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 6377
Answer
n = 6377
11.Hypothesis tests & confidence intervals
Answer: t = −1.799
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 = 20/√49 = 20/7 = 2.857
Test statistic
t = (69.86 − 75)/2.857 = −1.799
Answer
t = −1.799
t0.025; df = 48 = 2.011
|t| < 2.011, so H₀ is not rejected at the 5% level (two-sided).
12.Hypothesis tests & confidence intervals
Answer: U = 49.44
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 = 1.96·8/√81 = 1.742
Interval limit
U = 47.7 + 1.742 = 49.44
Answer
U = 49.44
13.Linear regression & correlation
Answer: r = 0.7916
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 = 351/6 = 58.5
Sums of squares and products
Sxx = Σ(x − x̄)² = 17.5
Sxy = Σ(x − x̄)(y − ȳ) = 36.5
Syy = Σ(y − ȳ)² = 121.5
Correlation coefficient
r = 36.5/√(17.5·121.5) = 0.7916
Answer
r = 0.7916
14.Linear regression & correlation
Answer: x = 7.421
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 = 33/6 = 5.5
ȳ = 300/6 = 50
Sums of squares and products
Sxx = Σ(x − x̄)² = 17.5
Sxy = Σ(x − x̄)(y − ȳ) = 82
Regression line
b = 82/17.5 = 4.686
a = 50 − 4.686·5.5 = 24.23
ŷ = 24.23 + 4.686·x
Solve for x
x = (59 − 24.23)/4.686 = 7.421
Answer
x = 7.421
15.Chi-square tests
Answer: χ² = 24.38
Chi-square tests: compare observed counts O with the counts E expected under H₀ using χ² = Σ(O − E)²/E. Goodness of fit: E = n·p for each category, df = k − 1 (minus one more for each parameter estimated from the data). Independence in an r × c table: E = row total × column total / grand total, df = (r − 1)(c − 1). Reject H₀ when χ² exceeds the critical value for the chosen α.
Expected counts
day shift: 100·60/200 = 30, 100·60/200 = 30, 100·80/200 = 40
night shift: 100·60/200 = 30, 100·60/200 = 30, 100·80/200 = 40
Chi-square statistic
χ² = (23 − 30)²/30 + (20 − 30)²/30 + (57 − 40)²/40 + (37 − 30)²/30 + (40 − 30)²/30 + (23 − 40)²/40
χ² = 24.38
Answer
χ² = 24.38
df = 2. At α = 0.05 the critical value is 5.991, so H₀ (independence) is rejected.
16.Chi-square tests
Answer: χ²crit = 23.21
Chi-square tests: compare observed counts O with the counts E expected under H₀ using χ² = Σ(O − E)²/E. Goodness of fit: E = n·p for each category, df = k − 1 (minus one more for each parameter estimated from the data). Independence in an r × c table: E = row total × column total / grand total, df = (r − 1)(c − 1). Reject H₀ when χ² exceeds the critical value for the chosen α.
Degrees of freedom
df = (2 − 1)·(11 − 1) = 10
Table
χ²crit(df = 10, α = 0.01) = 23.209
Answer
χ²crit = 23.209
Reject H₀ if the computed χ² is larger than this value.
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