Quantitative Methods
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06.02 | Chapter 6: Assumption Checks

Read the Shape, Not a Perfect Bell

Shapiro-Wilk gives the first normality signal. Histograms add the visual evidence that helps you judge whether a group is only a little uneven or seriously distorted.

By the end of this lesson, you can

  • explain why histograms support rather than replace Shapiro-Wilk;
  • inspect balance, outliers, and clustering within each group;
  • interpret all three program histograms with the same routine; and
  • decide what to do when the test and the graph do not agree.

Visual normality check

What a histogram adds

The formal test tells you whether there is statistical evidence of non-normality. The histogram shows what may be causing the signal and whether the shape is severe enough to make the mean a poor summary.

Novice rule: Real samples rarely form a perfectly smooth bell. Look for major distortion, not small gaps or ordinary bumps.

1
BalanceIs the shape roughly balanced, or is it pulled strongly to one side?
2
OutliersAre any scores isolated far away from the main group of values?
3
ClusteringIs there one main hill, or are there two clearly separate peaks?

Keep the target clear

For ANCOVA, you inspect the dependent variable within each group. In this study, the dependent variable is PostTest and the groups are the three enrichment programs.

Walk with me

Histogram 1: After-School Tutoring

Name what you are looking at before interpreting it: PostTest scores for the After-School Tutoring group.

One group, one visual check

Read the whole shape before focusing on any single bar.

Mean80.5
SD5.287
N20
Histogram of PostTest scores for the After-School Tutoring group. The bars form one main cluster around the low eighties with no isolated extreme bar.
BalanceNo extreme pull to one side.
OutliersNo isolated score appears to drive the shape.
ClusteringOne main cluster is visible.

Decision anchor: Because the earlier Shapiro-Wilk result was acceptable and this histogram shows no major distortion, it is reasonable to treat this group’s PostTest scores as approximately normal.

Now you try

Use the same script for the other groups

Do not invent a new standard for each picture. Read both histograms with the same balance, outliers, and clustering questions.

Saturday Academy

Mean = 82.4, SD = 5.679, N = 20

Histogram of PostTest scores for the Saturday Academy group. The distribution has one broad cluster and no single isolated extreme score.

Summer Bridge

Mean = 86.45, SD = 5.539, N = 20

Histogram of PostTest scores for the Summer Bridge group. The distribution is mildly uneven but has one main cluster without an isolated extreme score.

Quick check

What is the best beginner conclusion?

Use the graphs as visual evidence, not as a perfection contest.

Select one answer.

Research judgment

When Shapiro-Wilk and the histogram do not match

A p-value and a graph answer related but different questions. When they point in different directions, inspect the severity instead of following either one mechanically.

Significant test, mild graphProceed cautiouslyIf p ≤ .05 but the histogram shows only minor unevenness, you may continue cautiously and report the concern.
Nonsignificant test, severe graphPause and investigateIf p > .05 but the histogram shows extreme skew, strong outliers, or multiple peaks, investigate further.

Decision check

Which evidence should make you pause?

A group has Shapiro-Wilk p = .19, but its histogram has one extreme score far from every other value. What is the best next step?

Make it real

Explain one visual warning sign

Choose extreme skew, outliers, or multiple peaks. Explain why it could make a group mean less representative.

Sentence starter:

I would be most concerned about ______ because ______. This matters for ANCOVA because the adjusted comparison depends on group means that ______.

Lesson 06.02 | Source boundary: printed workbook pages 605–609. The lesson ends before Assumption 3: Homogeneity of Variance (Levene’s Test).