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
Demand characteristics are:
- AThe formal demands an ethics committee makes before a study can begin
- BCharacteristics of the sample
- CStatistical requirements for a test
- DCues that make participants guess the aim and change their behaviour
2.Research methods in psychologyHard
A study finds that students who sleep more have higher grades. Which design change would best allow a causal conclusion?
- ASurvey more students
- BRandomly assign students to different sleep schedules
- CMeasure grades more accurately
- DAsk students to report their sleep twice and average the two reports
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 53, 40, 33, 14, 25.
n = 5
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 50 and standard deviation 15. What proportion P of values is above 35?
μ = 50σ = 15x = 35
P =
5.ProbabilityMedium
In how many ways N can a committee of 5 people be chosen from a group of 13? The order does not matter.
n = 13k = 5
N =
6.ProbabilityHard
12 components are tested, and each one fails with probability 0.5, independently. What is the probability P that at most one fails?
n = 12p = 0.5
P =
7.Normal distributionMedium
Drag z so that the shaded area between −z and z under the standard normal curve is 0.9756. Find z to two decimals.
A = 0.9756
z =
8.Normal distributionHard
The heights of adult men are normally distributed with mean 178 cm and standard deviation 7 cm. Drag the cut-off so the shaded top 20% is to the right of it. Find the cut-off x.
μ = 178 cmσ = 7 cm20%
x = cm
9.Confidence intervalsMedium
A sample of 20 items has mean 97.5 and standard deviation s = 15. The population is roughly normal. Find the lower limit of the 99% confidence interval for the mean.
n = 20x̄ = 97.5s = 1599%t(19) = 2.861
lower limit =
10.Confidence intervalsHard
Nothing is known in advance about the proportion of defective products. How many products n must be inspected so that a 99% confidence interval for the proportion has a margin of error of at most 0.02?
p = 0.5E = 0.0299%
n =
11.Hypothesis tests & confidence intervalsMedium
A sample of 100 students has a mean score of 73.92 and a sample standard deviation of 12. Find the test statistic t for H₀: μ = 75.
μ₀ = 75x̄ = 73.92s = 12n = 100
t =
12.Hypothesis tests & confidence intervalsHard
A sample of 81 measurements has mean 119.3. The population standard deviation is 15. Find the lower limit L of the 90% confidence interval for the mean.
x̄ = 119.3σ = 15n = 8190%
L =
13.Linear regression & correlationMedium
The table gives 6 paired observations of x = advertising spend in k€ and y = sales in k€. Their sums are given. Find Pearson's correlation coefficient r from these sums.
n = 6Σx = 33Σy = 379Σx² = 199Σy² = 24431Σxy = 2175
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 “urban, medium”.
row total = 78column total = 62N = 178
E =
16.Chi-square testsHard
A chi-square goodness-of-fit test compares observed counts in 10 categories with fully specified expected proportions (no parameters are estimated from the data). How many degrees of freedom df does the test have?
k = 10
df =
3Solutions
1.Research methods in psychology
Answer: D · Cues that make participants guess the aim and change their behaviour
Why this is the answer
When participants work out the hypothesis they may act to confirm or sabotage it, threatening validity. Deception and blind procedures reduce this.
Answer
D · Cues that make participants guess the aim and change their behaviour
2.Research methods in psychology
Answer: B · Randomly assign students to different sleep schedules
Why this is the answer
Only manipulating sleep with random allocation controls third variables and allows causal inference, within ethical limits.
Answer
B · Randomly assign students to different sleep schedules
3.Descriptive statistics
Answer: s = 14.78
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 = 165/5 = 33
Squared deviations
Σ(x − x̄)² = 20² + 7² + 0² + (−19)² + (−8)² = 874
Standard deviation
s = √(874/4) = 14.78
Answer
s = 14.78
4.Descriptive statistics
Answer: P = 0.8413
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 = −1
Standard normal table
Φ(−1) = 0.1587
P = 1 − 0.1587 = 0.8413
Answer
P = 0.8413
5.Probability
Answer: N = 1287
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.
Combinations
C(13, 5) = (13·12·11·10·9)/(5·4·3·2·1) = 1287
Answer
N = 1287
6.Probability
Answer: P = 0.003174
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) = 0.512 = 0.0002441
P(X = 1) = 12·0.5·0.511 = 0.00293
Add them
P = 0.0002441 + 0.00293 = 0.003174
Answer
P = 0.003174
7.Normal distribution
Answer: z = 2.25
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 + 0.9756)/2 = 0.9878
Read the z-table backwards
Find 0.9878 inside the table and read off z.
z = 2.25
Answer
z = 2.25
8.Normal distribution
Answer: x = 183.9 cm
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
Φ(z) = 0.8
z from the table
z ≈ 0.84
Back to the original units
x = 178 + 0.84·7 = 183.9 cm
Answer
x = 183.9 cm
9.Confidence intervals
Answer: lower limit = 87.9
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 = 19, t = 2.861
Margin of error
E = 2.861·15/√20 = 9.596
Interval
97.5 ± 9.596 → [87.9, 107.1]
Answer
lower limit = 87.9
10.Confidence intervals
Answer: n = 4148
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 = 2.576²·0.5·0.5/0.02² = 4147
Round up
n = 4148
Answer
n = 4148
11.Hypothesis tests & confidence intervals
Answer: t = −0.9
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 = 12/√100 = 12/10 = 1.2
Test statistic
t = (73.92 − 75)/1.2 = −0.9
Answer
t = −0.9
t0.025; df = 99 = 1.984
|t| < 1.984, so H₀ is not rejected at the 5% level (two-sided).
12.Hypothesis tests & confidence intervals
Answer: L = 116.6
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/√81 = 2.742
Interval limit
L = 119.3 − 2.742 = 116.6
Answer
L = 116.6
13.Linear regression & correlation
Answer: r = 0.9765
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.
Centred sums
Sxy = 2175 − 33·379/6 = 90.5
Sxx = 199 − 33²/6 = 17.5
Syy = 24431 − 379²/6 = 490.8
Correlation coefficient
r = 90.5/√(17.5·490.8) = 0.9765
Answer
r = 0.9765
14.Linear regression & correlation
Answer: ŷ = 16.05
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 = 54/5 = 10.8
Sums of squares and products
Sxx = Σ(x − x̄)² = 40
Sxy = Σ(x − x̄)(y − ȳ) = 70
Regression line
b = 70/40 = 1.75
a = 10.8 − 1.75·6 = 0.3
ŷ = 0.3 + 1.75·x
Prediction
ŷ = 0.3 + 1.75·9 = 16.05
Answer
ŷ = 16.05
15.Chi-square tests
Answer: E = 27.17
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 = 78·62/178 = 27.17
Answer
E = 27.17
16.Chi-square tests
Answer: df = 9
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 = 10 − 1 = 9
Answer
df = 9
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