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Stat Dojo / Foundations

Graphical Analysis

Histograms, box plots, dot plots, run charts, and scatter plots: look at the data before you test it.

Question it answers
What does the data actually look like?
Data needed
Measurements, ideally with their order and group
Key output
Shape, clusters, outliers, trends, and shifts
Habit
Plot first, then summarize, then test
Assumptions
None; graphs are how you check assumptions
Excel
Insert > Statistic Chart (Histogram, Box and Whisker); Scatter; Line
Minitab
Graph > Histogram, Boxplot, Scatterplot; Stat > Quality Tools > Run Chart
Why it matters
A picture catches what a summary hides

The Idea in Plain Language

A graph shows what a number cannot: the shape, the clusters, the outliers, the trend, the shifts. Look at the data before you test it. A mean and a standard deviation can be identical for a healthy process and a broken one, and only a picture will tell them apart.

GraphShowsBest for
HistogramShape of the distribution of one variableSpread, skew, multiple peaks, comparison with the specification
Box plotMedian, quartiles, and outliersComparing several groups side by side
Dot plot / individual value plotEvery data pointSmall samples, where a box hides the detail
Scatter plotTwo variables against each otherRelationships, clusters, outliers
Run chart / time series plotValues in time orderTrends, shifts, cycles
Pareto chartCategories ranked by countFinding the vital few
Normal probability plotFit to the normal distributionChecking assumptions
Why it matters. Most analysis mistakes come from skipping the picture. The graph is the cheapest test and often the only one you need.

Worked Example 1: A Histogram Hides Nothing

Sixty shafts were measured. The summary looks harmless: mean 51.41 mm, median 51.65 mm, standard deviation 1.78 mm. A normality test gives p = 0.003.

  1. Choose the number of bins. Sturges’ rule suggests k = 1 + log2(60) = 7. Here, 1 mm bins from 48 to 55 are easy to read.
  2. Count the values in each bin, using the rule lower edge ≤ value < upper edge.
  3. Draw a bar for each bin; the height is the count.
Bin (mm)48–4949–5050–5151–5252–5353–5454–55
Count5139413142
0 2 4 6 8 10 12 14 16 48 49 50 51 52 53 54 55 Length (mm)
Two humps with a valley between them. The mean, which falls in the valley, describes almost no actual shaft.
47 48 49 50 51 52 53 54 55 56 Machine A Machine B Length (mm) box = middle 50%, line = median, diamond = mean, circle = outlier
Splitting by machine explains the two humps. Machine B runs about 3 mm longer than Machine A.
Conclusion. The data are a mixture of two machines (means 49.85 and 52.98 mm, Welch t = 14.8). The overall mean and standard deviation are meaningless here: no shaft is near 51.4 mm. The fix is to analyze the machines separately and find out why they differ.

Worked Example 2: A Run Chart Finds the Shift

Thirty daily averages in time order. The median is 104.0. A histogram of these values would look like one wide hump and hide the change entirely.

94 98 102 106 110 Median 103.95 14 in a row above the median Day Daily average
A run chart plots values in time order against the median. A long stretch on one side of the median signals a change.
  • The longest run on one side of the median is 14 points, starting on day 17. A run of 8 or more on one side of the median is unlikely to happen by chance.
  • The average of the first 14 days is 100.0, and the average of the last 16 days is 106.3.
Conclusion. The process shifted upward by about 6.2 units around day 15. A histogram of all 30 values would not reveal that. Ask what changed on day 15, and use a control chart for ongoing monitoring.

Scatter Plots and Choosing a Graph

Twenty-four parts were cured at different temperatures. The scatter plot shows a clear upward trend (r = 0.96); a table of numbers would not.

7.5 10 12.5 15 17.5 20 22.5 140 160 180 200 220 240 260 Cure temperature (C) Strength (MPa)
Each point is one part. Look for the direction, the strength, curvature, outliers, and separate clusters.
If you want to…Use
See the shape of one variableHistogram, or a dot plot for small samples
Compare several groupsBox plot or individual value plot by group
See change over timeRun chart; a control chart if you need limits
See a relationship between two measurementsScatter plot; a matrix plot for many pairs
Rank causes of defectsPareto chart
Check if data are normalProbability plot
Compare with specificationsHistogram with spec limits; capability plot

Run It in Excel and Minitab

ExcelStep by step

  1. Histogram: select the data, then Insert > Insert Statistic Chart > Histogram. Right-click the axis, choose Format Axis, and set the bin width. (The older route is Data > Data Analysis > Histogram.)
  2. Box plot: Insert > Insert Statistic Chart > Box and Whisker, one series per group. Excel’s box uses its own quartile method, so values can differ slightly from Minitab.
  3. Scatter plot: select two columns, then Insert > Scatter (X, Y). Add a trendline from the chart elements menu.
  4. Run chart: Insert > Line with Markers. Add a median line by adding a column of =MEDIAN($A$2:$A$31) and plotting it as a second series.
  5. Label every axis with its units, and give each chart a title that states the finding.

MinitabStep by step

  1. Graph > Histogram > Simple (add a fitted distribution under Data View if useful). To compare groups, use the With Groups or panel options.
  2. Graph > Boxplot > One Y, With Groups and Graph > Individual Value Plot.
  3. Graph > Scatterplot > With Regression and Graph > Matrix Plot for many variables.
  4. Stat > Quality Tools > Run Chart or Graph > Time Series Plot. The run chart reports tests for clustering, mixtures, trends, and oscillation.
  5. Stat > Basic Statistics > Graphical Summary gives a histogram, a box plot, and the main statistics in one view.
  6. Right-click any graph and choose Editor to change bins, scales, and labels.
Minitab session window: Descriptive Statistics beside a graphical summary (typed excerpt, simplified)
Descriptive Statistics: Length

Variable   N   Mean  StDev  Minimum     Q1  Median     Q3  Maximum
Length    60  51.41   1.78    48.20  49.90   51.65  53.00    54.40

Nothing in these numbers hints at a problem. The histogram shows two humps.

Reading and Reporting

  1. Look for shape: one hump or several? Symmetric or lopsided?
  2. Look for gaps and outliers, and find out what they are before removing them.
  3. Look over time: a trend, a shift, or a cycle means the process is not stable.
  4. Compare groups on the same scale, and label axes with units.
  5. Put the finding in the title: “Machine B runs 3 mm longer than Machine A,” not “Length by machine.”

Common Mistakes

MistakeWhy it misleadsBetter
Testing before plottingA mixture or outlier can invalidate the testAlways plot first
Too few or too many histogram binsHides shape or shows noiseTry Sturges’ rule, then a few more and fewer
Box plots on very small samplesQuartiles of 5 points are unstableUse an individual value plot
Plotting time-ordered data as a histogram onlyHides trends and shiftsAlso plot a run chart
Truncated or different axes between chartsExaggerates or hides differencesUse the same scale for comparisons
Pie charts and 3-D barsHard to compare accuratelyUse a sorted bar chart or Pareto

Try It Yourself

You have 40 measurements of a part from two suppliers, mixed in one column, and a histogram that looks flat-topped and wide.

  • What would you do first?
  • Which graph would you add to explain it?
Show the answer

Suspect a mixture of two populations. Split the data by supplier and compare them.

A box plot or individual value plot by supplier would show whether they are centered differently. Also plot the values in time order to check for shifts. Only then run any test, and analyze the suppliers separately if they differ.

Graphical Analysis: Frequently Asked Questions

Which graph should I use first?

For one variable, start with a histogram or a dot plot. For several groups, a box plot or individual value plot. For two variables, a scatter plot. For anything measured over time, a run chart.

How many bins should a histogram have?

Sturges’ rule, k = 1 + log2(n), is a start: 60 observations give about 7 bins. Try a few more and a few fewer. Too few hides the shape, too many shows noise.

What is the difference between a histogram and a bar chart?

A histogram shows the distribution of a continuous measurement, with touching bars for ranges. A bar chart compares separate categories, with gaps between the bars.

What do the parts of a box plot mean?

The box covers the middle 50% of the data (Q1 to Q3), the line is the median, the whiskers reach to the furthest values within 1.5 times the interquartile range, and the points beyond are outliers.

Why plot data in time order?

Most tests assume the process is stable. A run chart reveals trends, shifts, and cycles that a histogram or a summary will not show.

Sources and Further Reading

  • NIST/SEMATECH, e-Handbook of Statistical Methods, Exploratory Data Analysis (itl.nist.gov/div898/handbook).
  • John W. Tukey, Exploratory Data Analysis, Addison-Wesley.
  • Herbert A. Sturges, “The Choice of a Class Interval,” Journal of the American Statistical Association, 1926.
  • Minitab Support, “Graph menu” and “Run Chart” (support.minitab.com).

This content is educational. Worked examples use made-up data. Menu names for Minitab follow recent versions of Minitab Statistical Software and can differ slightly in older releases; Excel steps use Microsoft 365 and the Analysis ToolPak.