IB Psychology HLTopic 5 — Data AnalysisPaper 3 & IAHL only~12 min read
Probability and Inferential Statistics
Descriptive statistics tell you what happened in your sample. Inferential statistics answer the harder question: could that result have happened by chance alone? Everything in this section — hypotheses, significance levels, critical values — exists to answer that one question honestly.
📚 What you need to know
Research starts with a null hypothesis (H₀) predicting no difference or no relationship.
The alternative hypothesis (H₁) predicts the difference or relationship, and must be operationalised.
Directional (one-tailed) hypotheses predict which way the result will go; non-directional (two-tailed) ones do not.
The level of significance, written as p, is the probability the result happened by chance.
p < 0.05 is the standard threshold in psychology; p < 0.01 is used when there is a human cost or contradictory previous findings.
The observed value from your test is compared with a critical value from statistical tables.
A Type I error is a false positive; a Type II error is a false negative.
Why the null hypothesis comes first
All published psychological research must include a null hypothesis, and this is where every study starts. The null assumes the IV will not affect the DV. It is the default position, held until the evidence is strong enough to abandon it.
That may feel backwards, but it is deliberately cautious. Assuming nothing is happening until the data force you to conclude otherwise stops researchers finding patterns everywhere they look. The alternative hypothesis only gets accepted once the null has been rejected.
The two hypotheses
H₀: there will be no difference (or no relationship)
H₁: there will be a difference (or a relationship)
Both must be operationalised, meaning the IV and DV are written in measurable terms. “Caffeine improves memory” is not enough. “Participants who drink 200 ml of caffeine before a memory test will correctly recall more items out of 15 than participants who drink 200 ml of water” is a proper directional hypothesis. The non-directional version keeps the same detail but predicts only “there will be a difference”.
For correlational research, swap one word. Use “relationship” or “correlation” instead of “difference”. Everything else about writing the hypothesis stays exactly the same.
What p < 0.05 actually means
The significance level reflects how likely it is that chance factors are responsible for your results. At p < 0.05, you are accepting a probability of 5% or less that the result occurred by chance. That is the standard threshold in psychology — strict enough to be meaningful, loose enough to detect real effects.
Note that 5% is a choice, not a law of nature. Researchers move to 1% when a wrong conclusion would be costly — drug trials being the obvious case.
The stricter p < 0.01 level, meaning a 1% probability of chance, is used in two situations. First, when there is a potential human cost — a drug trial, where a false positive could put people at risk. Second, when previous research findings have been contradictory, so a stronger standard of evidence is needed before adding another claim.
“Significant” in statistics does not mean “big” or “important”. A tiny difference can be statistically significant with a large enough sample, and a large difference can fail to reach significance with a small one. Keep the everyday meaning out of your answer.
Using statistical tables
Once a statistical test is run, it produces an observed value, sometimes called the calculated value. On its own that number means nothing. It has to be compared with a critical value read from a table of critical values, and to find the right critical value you need three pieces of information.
🧩 Three questions before you read the table
Is the test one-tailed or two-tailed? In other words, does the alternative hypothesis predict a direction, or just that there will be a difference?
What is N? How many participants are in the sample, or how many degrees of freedom does the test have?
Which significance level applies? The standard is 0.05, unless there is a reason to be stricter.
Read off the critical value at the intersection of those three.
Compare it with your observed value and apply the rule the test specifies.
If the calculated value meets or exceeds the critical value, the null hypothesis can be rejected and the result is described as significant. Some tests reverse this rule — for a few, the observed value must be equal to or less than the critical value — so always check which direction applies to the test you are using.
Type I and Type II errors
Because the threshold is a probability rather than a certainty, mistakes are built into the system. There are exactly two ways to get it wrong.
Error
What the researcher does
Everyday name
More likely when
Type I
Rejects the null hypothesis when it should have been accepted
False positive — claiming an effect that is not there
The significance level is too lenient, such as 0.1 rather than 0.05
Type II
Accepts the null hypothesis when it should have been rejected
False negative — missing a real effect
The significance level is too strict, such as 0.01 rather than 0.05
Notice how the two pull in opposite directions. Loosen the threshold and you catch more real effects but also more imaginary ones. Tighten it and you avoid false alarms but start missing genuine findings. The 0.05 convention exists precisely because it guards reasonably against both.
A memory hook
Type I = you say there is something. There is not.
Type II = you say there is not. There is.
Worked examples
WORKED EXAMPLE
Write both hypotheses
A researcher predicts that students who revise in silence will score higher on a 30-mark test than students who revise with music. Write a suitable alternative hypothesis and null hypothesis, and state whether the test is one-tailed or two-tailed.
Step 1: Check whether a direction is predicted
“Will score higher” states a direction, so this is directional, or one-tailed.
Step 2: Operationalise the IV and DV
IV = revising in silence or with music. DV = score out of 30 on the test.
Step 3: Write H₁
Students who revise in silence will score significantly higher out of 30 than students who revise with music playing.
Step 4: Write H₀
There will be no significant difference in scores out of 30 between students who revise in silence and students who revise with music playing.
Directional H₁ plus a matching non-directional nullthe null is always non-directional — it just says “no difference”
WORKED EXAMPLE
Interpret the result and identify the error risk
A study finds a difference between conditions with p = 0.04. Explain what conclusion can be drawn, and which type of error remains possible.
Step 1: Compare with the threshold0.04 < 0.05, so the result is significant at the standard level.
Step 2: State the conclusion correctly
The null hypothesis is rejected and the alternative hypothesis is accepted. There is a 4% probability the result was due to chance.
Step 3: Identify the error still possible
A Type I error — a false positive. There remains a 4% chance the null was rejected when it should have been accepted.
Step 4: Add the caution
Significance says nothing about the size of the effect, only that it is unlikely to be chance. Replication would strengthen the claim.
Significant at p < 0.05; Type I error remains possiblerejecting the null always leaves a Type I risk; accepting it leaves a Type II risk
💡 Exam tip
Operationalise both variables in every hypothesis you write. Unmeasurable hypotheses lose the mark.
Say “reject the null hypothesis”, not “prove the alternative hypothesis”. Statistics never proves anything.
Give the two reasons for using p < 0.01: human cost, and contradictory previous findings.
Learn the three table questions: tailed or not, N, and significance level.
Keep Type I and Type II straight with the false positive and false negative labels.
State explicitly that statistical significance is not the same as practical importance.
⚠ Common mix-up
Thinking significant means large. It means unlikely to be due to chance, nothing more.
Writing that a study proved the hypothesis. Research supports or fails to support; it never proves.
Making the null hypothesis directional. The null always predicts no difference or no relationship.
Assuming p < 0.01 is simply better. It reduces Type I errors while raising the risk of Type II errors.
Swapping the two error types. Type I claims something that is not there; Type II misses something that is.
Forgetting the direction of the comparison rule. Some tests need the observed value to be lower than the critical value, not higher.
Up next: Choosing the Right Statistical Test — three questions that lead you to exactly one test, every time.
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