Read a histogram's shape, recognise skew, and flag outliers with the 1.5 × IQR rule.
Why this matters
A mean and a standard deviation compress a whole column into two numbers. That is useful, but it can hide a lot: two peaks, a long tail, a cluster of impossible values from a broken sensor. A histogram shows the whole distribution, meaning how often each range of values occurs. Looking at it before you summarise is one of the habits that separates careful analysts from careless ones.
Building a histogram
- Choose bins: equal-width ranges that cover the data, for example 0 to 10, 10 to 20, 20 to 30 minutes.
- Count how many values fall in each bin.
- Draw a bar for each bin, with height equal to the count. The bars touch, because the bins are continuous.
A text version of wait times (minutes) at a help desk:
0-10 ########## 10
10-20 ############ 12
20-30 ###### 6
30-40 ### 3
40-50 # 1
50-60 0
60-70 # 1
Bin width matters. Too few bins hide structure; too many turn the picture into noise. Start with something like 5 to 15 bins and try a couple of widths.
A histogram is not a bar chart. A bar chart shows categories (blood type, material) with gaps between bars, and the order of bars is up to you. A histogram shows a numerical variable on a number line.
Describing shape
Describe a distribution with four things: shape, centre, spread and unusual values.
Symmetric: the left and right halves look roughly alike. Repeated measurements of the same quantity and heights of adults of one sex are classic examples. Mean and median are close.
Right-skewed (positively skewed): a long tail to the right. Most values are small, a few are large. Wait times, incomes, file sizes, reaction times, city populations. The mean is pulled toward the tail, so mean > median.
Left-skewed (negatively skewed): a long tail to the left. Exam scores on an easy test, age at retirement, time to failure for components that rarely fail early. Mean < median.
Bimodal: two peaks. Often a sign that two groups are mixed together, such as morning and evening shifts, two machines, or two species. Splitting the data by group usually explains it.
The help-desk histogram above is right-skewed: the bulk is under 30 minutes, with a tail out to 60+.
Quartiles and the box plot
Quartiles split sorted data into four equal parts. Q1 is the 25th percentile, Q2 the median, Q3 the 75th percentile. The IQR is Q3 − Q1.
A box plot draws a box from Q1 to Q3 with a line at the median, and "whiskers" out to the most extreme values that are not outliers. Box plots are compact, which makes them good for comparing several groups side by side.
Flagging outliers: the 1.5 × IQR rule
A widely used rule (Tukey's fences):
lower fence = Q1 - 1.5 * IQR
upper fence = Q3 + 1.5 * IQR
Any value outside the fences is flagged as a possible outlier.
Worked example
Wait times: 4, 7, 8, 9, 11, 12, 14, 15, 18, 22, 65. Suppose Q1 = 8 and Q3 = 18 (software may give slightly different quartiles; there are several methods).
IQR = 18 - 8 = 10
lower fence = 8 - 15 = -7
upper fence = 18 + 15 = 33
65 is above 33, so it is flagged. Nothing is below −7.
"Flagged" does not mean "delete". An outlier can be a typing error (6.5 entered as 65), a broken instrument, or a real and important event (the one customer who waited an hour). Investigate first. Remove a value only with a documented reason, and say so in your report.
In Python
import statistics
waits = [4, 7, 8, 9, 11, 12, 14, 15, 18, 22, 65]
q1, q2, q3 = statistics.quantiles(waits, n=4)
print(q1, q2, q3) # 8.0 12.0 18.0
print(statistics.mean(waits), statistics.median(waits)) # mean 16.8 > median 12: right skew
statistics.quantiles(data, n=4) returns the three cut points between quartiles.
Common mistakes
- Summarising with the mean before looking at the shape.
- Using a histogram for categorical data, or a bar chart with gaps for continuous data.
- Deleting outliers automatically because a rule flagged them.
- Judging skew from one bin width only.
Quick recap
- A histogram shows how often values fall in equal-width bins.
- Describe shape (symmetric, right-skewed, left-skewed, bimodal), centre, spread and unusual values.
- Right skew: mean > median. Left skew: mean < median.
- Fences at Q1 − 1.5 × IQR and Q3 + 1.5 × IQR flag possible outliers; investigate before removing.
Try it
In the Do tab, finish an outlier detector by writing the two fence formulas.