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
Why are behaviour categories in an observation operationalised before data collection?
- ASo participants know exactly what is being recorded
- BSo the hypothesis can be changed after the data are seen
- CSo the observation becomes an experiment
- DSo observers record the same behaviours consistently, improving reliability
2.Research methods in psychologyHard
In grounded theory, data collection continues until:
- AA p-value below 0.05 is obtained
- BTheoretical saturation is reached and new data add no new categories
- CThe original hypothesis is confirmed
- DAt least 1,000 participants have been interviewed
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 14, 55, 13, 18, 55, 44, 35.
n = 7
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 170 and standard deviation 12. What proportion P of values is above 174.8?
μ = 170σ = 12x = 174.8
P =
5.ProbabilityMedium
A bag holds 7 red and 8 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?
7 / 8
P =
6.ProbabilityHard
A test has 10 multiple-choice questions. A student who guesses gets each one right with probability 0.3, independently. What is the probability P of exactly 1 correct answers?
n = 10p = 0.3k = 1
P =
7.Normal distributionMedium
Drag z so that the shaded area under the standard normal curve, to the right of z, is 0.0918. Find z to two decimals.
A = 0.0918
z =
8.Normal distributionHard
The exam scores are normally distributed with mean 65 points and standard deviation 10 points. Find the value x below which 97.5% of them lie (the 97.5th percentile).
μ = 65σ = 10P = 97.5%
x = points
9.Confidence intervalsMedium
A sample of 8 items has mean 94 and standard deviation s = 16. The population is roughly normal. Find the margin of error E of a 95% confidence interval for the mean.
n = 8x̄ = 94s = 1695%t(7) = 2.365
E =
10.Confidence intervalsHard
A pilot study suggests that about 0.1 of products are defective. How many products n must be inspected so that a 95% confidence interval for the proportion has a margin of error of at most 0.025?
p ≈ 0.1E = 0.02595%
n =
11.Hypothesis tests & confidence intervalsMedium
A sample of 81 students has a mean score of 121.2 and a sample standard deviation of 5. Find the test statistic t for H₀: μ = 120.
μ₀ = 120x̄ = 121.2s = 5n = 81
t =
12.Hypothesis tests & confidence intervalsHard
A sample of 81 measurements has mean 201.7. The population standard deviation is 12. Find the lower limit L of the 99% confidence interval for the mean.
x̄ = 201.7σ = 12n = 8199%
L =
13.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 =
14.Linear regression & correlationHard
The table gives 6 paired observations of x = load in kN and y = extension in mm. Use the least-squares line ŷ = a + b·x. Predict ŷ for x = 11.
n = 6x = 11
ŷ =
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 = 2402 × 3
χ² =
16.Chi-square testsHard
A chi-square goodness-of-fit test checks whether counts in 5 categories follow a Poisson distribution whose mean is estimated from the same data. Using the table, find the critical value χ²crit at significance level α = 0.05.
α = 0.05
χ²crit =
3Solutions
1.Research methods in psychology
Answer: D · So observers record the same behaviours consistently, improving reliability
Why this is the answer
Clear, observable categories (for example "hits with an open hand") reduce subjective judgement and raise inter-observer reliability.
Answer
D · So observers record the same behaviours consistently, improving reliability
2.Research methods in psychology
Answer: B · Theoretical saturation is reached and new data add no new categories
Why this is the answer
Grounded theory develops theory inductively, collecting and coding data together, and stops sampling when new data no longer develop the categories.
Answer
B · Theoretical saturation is reached and new data add no new categories
3.Descriptive statistics
Answer: s = 18.61
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 = 234/7 = 33.43
Squared deviations
Σ(x − x̄)² = (−19.43)² + 21.57² + (−20.43)² + (−15.43)² + 21.57² + 10.57² + 1.571² = 2078
Standard deviation
s = √(2078/6) = 18.61
Answer
s = 18.61
4.Descriptive statistics
Answer: P = 0.3446
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 = 0.4
Standard normal table
Φ(0.4) = 0.6554
P = 1 − 0.6554 = 0.3446
Answer
P = 0.3446
5.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 = 7/15 · 6/14 = 42/210 = 0.2
Answer
P = 0.2
6.Probability
Answer: P = 0.1211
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 = 1) = 10·0.31·0.79 = 10·0.3·0.04035 = 0.1211
Answer
P = 0.1211
7.Normal distribution
Answer: z = 1.33
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 − 0.0918 = 0.9082
Read the z-table backwards
Find 0.9082 inside the table and read off z.
z = 1.33
Answer
z = 1.33
8.Normal distribution
Answer: x = 84.6 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) = 0.975 → z = 1.96
Back to x
x = 65 + 1.96·10 = 84.6 points
Answer
x = 84.6 points
9.Confidence intervals
Answer: E = 13.38
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 = 7, t = 2.365
Margin of error
E = 2.365·16/√8 = 13.38
Answer
E = 13.38
10.Confidence intervals
Answer: n = 554
Confidence interval for a proportion: p̂ ± z·√(p̂(1 − p̂)/n), where p̂ = x/n. To reach a margin E you need n ≥ z²·p(1 − p)/E², rounded up; without a prior estimate use p = 0.5, which gives the largest n.
Solve for n
n = 1.96²·0.1·0.9/0.025² = 553.2
Round up
n = 554
Answer
n = 554
11.Hypothesis tests & confidence intervals
Answer: t = 2.16
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 = 5/√81 = 5/9 = 0.5556
Test statistic
t = (121.2 − 120)/0.5556 = 2.16
Answer
t = 2.16
t0.025; df = 80 = 1.99
|t| > 1.99, so H₀ is rejected at the 5% level (two-sided).
12.Hypothesis tests & confidence intervals
Answer: L = 198.3
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 = 2.576·12/√81 = 3.435
Interval limit
L = 201.7 − 3.435 = 198.3
Answer
L = 198.3
13.Linear regression & correlation
Answer: r = 0.9866
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 = 223/5 = 44.6
Sums of squares and products
Sxx = Σ(x − x̄)² = 10
Sxy = Σ(x − x̄)(y − ȳ) = 32
Syy = Σ(y − ȳ)² = 105.2
Correlation coefficient
r = 32/√(10·105.2) = 0.9866
Answer
r = 0.9866
14.Linear regression & correlation
Answer: ŷ = 32.06
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 = 131/6 = 21.83
Sums of squares and products
Sxx = Σ(x − x̄)² = 70
Sxy = Σ(x − x̄)(y − ȳ) = 179
Regression line
b = 179/70 = 2.557
a = 21.83 − 2.557·7 = 3.933
ŷ = 3.933 + 2.557·x
Prediction
ŷ = 3.933 + 2.557·11 = 32.06
Answer
ŷ = 32.06
15.Chi-square tests
Answer: χ² = 8.472
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
students: 120·48/240 = 24, 120·72/240 = 36, 120·120/240 = 60
staff: 120·48/240 = 24, 120·72/240 = 36, 120·120/240 = 60
Chi-square statistic
χ² = (19 − 24)²/24 + (46 − 36)²/36 + (55 − 60)²/60 + (29 − 24)²/24 + (26 − 36)²/36 + (65 − 60)²/60
χ² = 8.472
Answer
χ² = 8.472
df = 2. At α = 0.05 the critical value is 5.991, so H₀ (independence) is rejected.
16.Chi-square tests
Answer: χ²crit = 7.815
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 = 5 − 1 − 1 = 3
Table
χ²crit(df = 3, α = 0.05) = 7.815
Answer
χ²crit = 7.815
Reject H₀ if the computed χ² is larger than this value.
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