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
A researcher studies the effect of gender on memory scores. Why is this a quasi-experiment rather than a true experiment?
- AThere are only two conditions, and a true experiment needs at least three levels of the IV
- BThe IV is a pre-existing characteristic, so participants cannot be randomly allocated to it
- CThe DV is not measured
- DThe study takes place in a laboratory
2.Research methods in psychologyHard
Which non-parametric test compares two related samples, such as before-and-after scores on an ordinal scale?
- AMann–Whitney U test
- BChi-square test of independence
- CIndependent t-test
- DWilcoxon signed-rank test
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 17, 11, 30, 46, 34, 35, 58.
n = 7
s =
4.Descriptive statisticsHard
Exam scores are normally distributed with mean 100 and standard deviation 8. Find the z-score z of a score of 93.6.
μ = 100σ = 8x = 93.6
z =
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 11 multiple-choice questions. A student who guesses gets each one right with probability 0.3, independently. What is the probability P of exactly 2 correct answers?
n = 11p = 0.3k = 2
P =
7.Normal distributionMedium
The exam scores are normally distributed with mean 65 points and standard deviation 10 points. What fraction lies between 53.5 and 83 points? Give P as a decimal.
μ = 65σ = 10x1 = 53.5x2 = 83
P =
8.Normal distributionHard
The heights of adult men are normally distributed with mean 178 cm and standard deviation 7 cm. Find the value x below which 95% of them lie (the 95th percentile).
μ = 178σ = 7P = 95%
x = cm
9.Confidence intervalsMedium
The blood pressure of 15 patients is measured before and after a treatment. The differences d = before − after (mmHg) have mean 2.8 and standard deviation 2.8. Find the margin of error E of the 95% confidence interval for the mean difference.
n = 15d̄ = 2.8sd = 2.895%t(14) = 2.145
E = mmHg
10.Confidence intervalsHard
The population standard deviation is 26. 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 = 26E = 390%
n =
11.Hypothesis tests & confidence intervalsMedium
Two production lines make the same part. Line 1: a sample of 100 parts has mean length 93.5 mm, and the line's known standard deviation is 8 mm. Line 2: a sample of 30 parts has mean 94.4 mm, known standard deviation 4 mm. Find the test statistic z for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (line 1 minus line 2).
n₁ = 100x̄₁ = 93.5 mmσ₁ = 8 mmn₂ = 30x̄₂ = 94.4 mmσ₂ = 4 mm
z =
12.Hypothesis tests & confidence intervalsHard
A sample of 100 measurements has mean 117.5. The population standard deviation is 15. Find the upper limit U of the 90% confidence interval for the mean.
x̄ = 117.5σ = 15n = 10090%
U =
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 5 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 = 9.
n = 5x = 9
ŷ =
15.Chi-square testsMedium
The contingency table shows observed counts. For a chi-square test of independence, find the expected count E for the cell “students, bus”.
row total = 85column total = 63N = 173
E =
16.Chi-square testsHard
A chi-square goodness-of-fit test checks whether counts in 6 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: B · The IV is a pre-existing characteristic, so participants cannot be randomly allocated to it
Why this is the answer
In a quasi-experiment the IV is a naturally occurring difference between people. Without random allocation, other differences between groups may confound the results.
Answer
B · The IV is a pre-existing characteristic, so participants cannot be randomly allocated to it
2.Research methods in psychology
Answer: D · Wilcoxon signed-rank test
Why this is the answer
The Wilcoxon signed-rank test is for paired ordinal data; Mann–Whitney U is its counterpart for independent groups.
Answer
D · Wilcoxon signed-rank test
3.Descriptive statistics
Answer: s = 16.06
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 = 231/7 = 33
Squared deviations
Σ(x − x̄)² = (−16)² + (−22)² + (−3)² + 13² + 1² + 2² + 25² = 1548
Standard deviation
s = √(1548/6) = 16.06
Answer
s = 16.06
4.Descriptive statistics
Answer: z = −0.8
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 = (93.6 − 100)/8 = −0.8
Answer
z = −0.8
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 = 0.1998
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 = 2) = 55·0.32·0.79 = 55·0.09·0.04035 = 0.1998
Answer
P = 0.1998
7.Normal distribution
Answer: P = 0.839
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 = (53.5 − 65)/10 = −1.15
z₂ = (83 − 65)/10 = 1.8
Area between
P = Φ(1.8) − Φ(−1.15) = 0.9641 − 0.1251 = 0.839
Answer
P = 0.839
8.Normal distribution
Answer: x = 189.5 cm
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) = 0.95 → z = 1.645
Back to x
x = 178 + 1.645·7 = 189.51 cm
Answer
x = 189.51 cm
9.Confidence intervals
Answer: E = 1.551 mmHg
Interval for a difference: estimate ± critical value × standard error. For two independent means with known σ: SE = √(σ₁²/n₁ + σ₂²/n₂) and z. With equal unknown variances: pooled s_p and t with n₁ + n₂ − 2 degrees of freedom. For paired data, work with the differences: d̄ ± t·s_d/√n.
Critical value
df = 14, t = 2.145
Margin of error
E = 2.145·2.8/√15 = 1.551
Answer
E = 1.551 mmHg
10.Confidence intervals
Answer: n = 204
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 = (1.645·26/3)² = 203.3
Round up
A sample size is a whole number, and rounding down would make the margin too large.
n = 204
Answer
n = 204
11.Hypothesis tests & confidence intervals
Answer: z = −0.8309
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.
Standard error of the difference
SE = √(8²/100 + 4²/30) = √1.173 = 1.083
Test statistic
z = (93.5 − 94.4)/1.083 = −0.8309
Answer
z = −0.8309
|z| < 1.96, so H₀ is not rejected at the 5% level (two-sided).
12.Hypothesis tests & confidence intervals
Answer: U = 120
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 = 1.645·15/√100 = 2.468
Interval limit
U = 117.5 + 2.468 = 120
Answer
U = 120
13.Linear regression & correlation
Answer: r = 0.9193
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 = 20/5 = 4
ȳ = 281/5 = 56.2
Sums of squares and products
Sxx = Σ(x − x̄)² = 10
Sxy = Σ(x − x̄)(y − ȳ) = 34
Syy = Σ(y − ȳ)² = 136.8
Correlation coefficient
r = 34/√(10·136.8) = 0.9193
Answer
r = 0.9193
14.Linear regression & correlation
Answer: ŷ = 23.4
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 = 30/5 = 6
ȳ = 78/5 = 15.6
Sums of squares and products
Sxx = Σ(x − x̄)² = 40
Sxy = Σ(x − x̄)(y − ȳ) = 104
Regression line
b = 104/40 = 2.6
a = 15.6 − 2.6·6 = 0
ŷ = 0 + 2.6·x
Prediction
ŷ = 0 + 2.6·9 = 23.4
Answer
ŷ = 23.4
15.Chi-square tests
Answer: E = 30.95
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 count
E = 85·63/173 = 30.95
Answer
E = 30.95
16.Chi-square tests
Answer: χ²crit = 9.488
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 = 6 − 1 − 1 = 4
Table
χ²crit(df = 4, α = 0.05) = 9.488
Answer
χ²crit = 9.488
Reject H₀ if the computed χ² is larger than this value.
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