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
Which design uses different participants in each condition?
- AIndependent groups design
- BRepeated measures design
- CMatched pairs design with the same people
- DWithin-subjects design
2.Research methods in psychologyHard
A researcher records data at nine time points and tests at the 5% level each time without correction. What is the problem?
- AThe data become more reliable each time, so the final test is the most accurate
- BThere is no problem with this approach
- CThe effect size falls with each test
- DThe chance of at least one false positive rises well above 5%
3.Descriptive statisticsMedium
Find the sample standard deviation s of the data: 60, 45, 27, 45, 14.
n = 5
s =
4.Descriptive statisticsHard
A quantity is normally distributed with mean 70 and standard deviation 20. What proportion P of values is above 110?
μ = 70σ = 20x = 110
P =
5.ProbabilityMedium
A bag holds 5 red and 5 blue balls. You draw two balls without putting the first one back. What is the probability P that both are red?
5 / 5
P =
6.ProbabilityHard
A test has 11 multiple-choice questions. A student who guesses gets each one right with probability 0.4, independently. What is the probability P of exactly 2 correct answers?
n = 11p = 0.4k = 2
P =
7.Normal distributionMedium
The masses of passengers with luggage are independent and normal with mean 85 kg and standard deviation 12 kg. Find the probability P that the total mass of 5 of them is below 451.8 kg. Round z to two decimals.
μ = 85 kgσ = 12 kgn = 5< 451.8 kg
P =
8.Normal distributionHard
The bag weights are normally distributed with mean 250 g and standard deviation 5 g. Drag the cut-off so the shaded bottom 20% is to the left of it. Find the cut-off x.
μ = 250 gσ = 5 g20%
x = g
9.Confidence intervalsMedium
A sample of 20 items has mean 53.5 and standard deviation s = 5. The population is roughly normal. Find the lower limit of the 99% confidence interval for the mean.
n = 20x̄ = 53.5s = 599%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 95% confidence interval for the proportion has a margin of error of at most 0.03?
p = 0.5E = 0.0395%
n =
11.Hypothesis tests & confidence intervalsMedium
A sample of 81 students has a mean score of 196.6 and a sample standard deviation of 10. Find the test statistic t for H₀: μ = 200.
μ₀ = 200x̄ = 196.6s = 10n = 81
t =
12.Hypothesis tests & confidence intervalsHard
A one-sided z test of H₀: μ = 80 against H₁: μ > 80 is done at α = 0.10 with a sample of 36; the standard deviation is known to be 18. In reality the mean is 87.8. Find the power 1 − β of the test in percent.
μ₀ = 80μ = 87.8σ = 18n = 36α = 0.10
1 − β = %
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 = temperature in °C and y = ice creams sold. Use the least-squares line ŷ = a + b·x. For which value x does the line predict ŷ = 50?
n = 6ŷ = 50
x =
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 “day shift, no defect”.
row total = 45column total = 60N = 162
E =
16.Chi-square testsHard
A chi-square goodness-of-fit test checks whether counts in 8 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: A · Independent groups design
Why this is the answer
In an independent groups design each condition has different participants, avoiding order effects but risking participant variables.
Answer
A · Independent groups design
2.Research methods in psychology
Answer: D · The chance of at least one false positive rises well above 5%
Why this is the answer
With nine independent tests the chance of at least one Type I error is about 1 − 0.95⁹ ≈ 37%. Corrections such as Bonferroni address this.
Answer
D · The chance of at least one false positive rises well above 5%
3.Descriptive statistics
Answer: s = 17.88
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 = 191/5 = 38.2
Squared deviations
Σ(x − x̄)² = 21.8² + 6.8² + (−11.2)² + 6.8² + (−24.2)² = 1279
Standard deviation
s = √(1279/4) = 17.88
Answer
s = 17.88
4.Descriptive statistics
Answer: P = 0.02275
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 = 2
Standard normal table
Φ(2) = 0.9772
P = 1 − 0.9772 = 0.02275
Answer
P = 0.02275
5.Probability
Answer: P = 0.2222
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 = 5/10 · 4/9 = 20/90 = 0.2222
Answer
P = 0.2222
6.Probability
Answer: P = 0.08868
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) = 55·0.42·0.69 = 55·0.16·0.01008 = 0.08868
Answer
P = 0.08868
7.Normal distribution
Answer: P = 0.8413
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 total
μT = 425 kg, σT = 12·√5 = 26.83 kg
Standardise
z = (451.8 − 425)/26.83 = 1
Probability
P = Φ(1) = 0.8413
Answer
P = 0.8413
8.Normal distribution
Answer: x = 245.8 g
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) = 0.2
z from the table
z ≈ −0.84
Back to the original units
x = 250 − 0.84·5 = 245.8 g
Answer
x = 245.8 g
9.Confidence intervals
Answer: lower limit = 50.3
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 = 19, t = 2.861
Margin of error
E = 2.861·5/√20 = 3.199
Interval
53.5 ± 3.199 → [50.3, 56.7]
Answer
lower limit = 50.3
10.Confidence intervals
Answer: n = 1068
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 = 1.96²·0.5·0.5/0.03² = 1067
Round up
n = 1068
Answer
n = 1068
11.Hypothesis tests & confidence intervals
Answer: t = −3.06
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 = 10/√81 = 10/9 = 1.111
Test statistic
t = (196.6 − 200)/1.111 = −3.06
Answer
t = −3.06
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: 1 − β = 90.62 %
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 = 18/√36 = 3
x̄c = 80 + 1.282·3 = 83.85
z under the true mean
z = (83.85 − 87.8)/3 = −1.318
Power
power = P(Z > −1.318) = 90.62%
1 − β = 90.62%
Answer
1 − β = 90.62%
13.Linear regression & correlation
Answer: r = 0.9983
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 = 25/5 = 5
ȳ = 248/5 = 49.6
Sums of squares and products
Sxx = Σ(x − x̄)² = 10
Sxy = Σ(x − x̄)(y − ȳ) = 60
Syy = Σ(y − ȳ)² = 361.2
Correlation coefficient
r = 60/√(10·361.2) = 0.9983
Answer
r = 0.9983
14.Linear regression & correlation
Answer: x = 18.92
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 = 138/6 = 23
ȳ = 387/6 = 64.5
Sums of squares and products
Sxx = Σ(x − x̄)² = 70
Sxy = Σ(x − x̄)(y − ȳ) = 249
Regression line
b = 249/70 = 3.557
a = 64.5 − 3.557·23 = −17.31
ŷ = −17.31 + 3.557·x
Solve for x
x = (50 − (−17.31))/3.557 = 18.92
Answer
x = 18.92
15.Chi-square tests
Answer: E = 16.67
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 = 45·60/162 = 16.67
Answer
E = 16.67
16.Chi-square tests
Answer: χ²crit = 12.59
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 = 8 − 1 − 1 = 6
Table
χ²crit(df = 6, α = 0.05) = 12.592
Answer
χ²crit = 12.592
Reject H₀ if the computed χ² is larger than this value.
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