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Data
Displays

Learn to represent numerical data with dot plots, histograms, and box plots: and choose the right display for any data set.

Pre-Algebra

What Is a Data Display?

A data display is a visual representation of numerical data. Instead of staring at a long list of numbers, we organize them into a picture that reveals patterns, clusters, gaps, and outliers at a glance.

Dot Plot
Shows every individual data value as a dot on a number line.
Histogram
Groups data into intervals (bins) and uses bars to show frequency.
Box Plot
Summarizes data with the five-number summary: min, \(Q_1\), median, \(Q_3\), max.

The Dot Plot

A dot plot places one dot above a number line for each data value. Stacked dots show how often a value appears. It is best for small data sets (roughly \( n \leq 30 \)) where you want to see every individual value.

\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
\(8\)
mode = \(5\)
Hours of Sleep (\(14\) students)

The Histogram

A histogram groups data into equal-width intervals (also called bins) and draws a bar whose height shows the frequency (count) for each interval. There are no gaps between bars because the intervals are continuous.

Test Scores (\(30\) students)
\(0\)
\(2\)
\(4\)
\(6\)
\(8\)
\(10\)
\(2\)
\(5\)
\(10\)
\(8\)
\(5\)
\(50\)-\(60\)
\(60\)-\(70\)
\(70\)-\(80\)
\(80\)-\(90\)
\(90\)-\(100\)
Frequency
Score Range

The Box Plot

A box plot (box-and-whisker plot) uses the five-number summary to display data spread. It highlights the median, quartiles, and any potential outliers.

Daily Temperatures (\(^{\circ}\text{F}\))
\(40\)
\(50\)
\(60\)
\(70\)
\(80\)
\(90\)
\(100\)
\(\min\)
\(45\)
\(Q_1\)
\(58\)
\(\text{Med}\)
\(72\)
\(Q_3\)
\(85\)
\(\max\)
\(96\)

The Five-Number Summary

Every box plot is built from these five statistics, calculated from the ordered data set:

1
Minimum: the smallest value in the data set.
2
\(Q_1\) (first quartile): the median of the lower half of the data.
3
Median (\(Q_2\)): the middle value of the entire data set.
4
\(Q_3\) (third quartile): the median of the upper half of the data.
5
Maximum: the largest value in the data set.

Dot Plot vs. Histogram vs. Box Plot

Dot Plot Histogram Box Plot
Shows individual values? Yes No (grouped) No (summary)
Best data set size Small (\( n \leq 30 \)) Medium to large Any size
Shows shape? Yes Yes Partially
Shows median/quartiles? Not directly Not directly Yes
Good for comparing groups? Possible Possible Excellent

How to Choose a Display

1
Count your data points. Fewer than ~\(30\)? A dot plot works well. More? Consider a histogram or box plot.
2
Decide what you want to show. Individual values → dot plot. Frequency distribution → histogram. Spread and quartiles → box plot.
3
Are you comparing groups? Side-by-side box plots make it easy to compare medians, ranges, and IQRs across two or more groups.
4
Check for outliers. A box plot flags outliers automatically. A dot plot shows them visually as isolated dots.
5
Label everything. Title, axis labels, units, and scale are essential for any display to make sense.

Example 1: Building a Dot Plot

Data: Number of pets owned by \(12\) students: \(0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 7\)

Number of Pets
\(0\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
mode = \(2\)
gap at \(6\)

Example 2: Building a Histogram

Data: Ages of \(20\) people at a park: \(5, 8, 12, 14, 15, 18, 20, 22, 24, 25, 27, 28, 30, 32, 35, 38, 40, 45, 55, 60\). Use bins of width \(10\).

Ages at the Park
\(0\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(2\)
\(4\)
\(6\)
\(4\)
\(2\)
\(1\)
\(1\)
\(0\)-\(10\)
\(10\)-\(20\)
\(20\)-\(30\)
\(30\)-\(40\)
\(40\)-\(50\)
\(50\)-\(60\)
\(60\)-\(70\)
Frequency
Age Range

Example 3: Building a Box Plot

Data (ordered): \(12, 15, 18, 22, 25, 28, 30, 33, 37, 42\). Find the five-number summary.

Min
\( 12 \)
\(Q_1\)
\( 18 \)
Median
\( \frac{25+28}{2} = 26.5 \)
\(Q_3\)
\( 33 \)
Max
\( 42 \)
\(10\)
\(15\)
\(20\)
\(25\)
\(30\)
\(35\)
\(40\)
\(45\)
\(12\)
\(18\)
\(26.5\)
\(33\)
\(42\)

Distribution Shapes

When you look at any data display, describe its shape using one of these terms:

Symmetric
Both halves are roughly mirror images. The mean and median are close together.
Skewed Right
A long tail stretches to the right. The mean is pulled higher than the median.
Skewed Left
A long tail stretches to the left. The mean is pulled lower than the median.

Identifying Outliers

An outlier is a data value that is unusually far from the rest of the data. You can identify them using the \(1.5 \times \text{IQR}\) rule:

1
Calculate \( \text{IQR} = Q_3 - Q_1 \).
2
Find the lower fence: \( Q_1 - 1.5 \times \text{IQR} \).
3
Find the upper fence: \( Q_3 + 1.5 \times \text{IQR} \).
4
Any data value below the lower fence or above the upper fence is an outlier.

Comparing with Side-by-Side Box Plots

Side-by-side box plots let you compare two (or more) data sets on the same scale.

Quiz Scores: Class A vs. Class B
\(50\)
\(60\)
\(70\)
\(80\)
\(90\)
\(100\)
A
B

Common Mistakes

✕ Gaps in Histograms
Leaving gaps between bars. Histogram bars must touch because the data is continuous.
✕ Unequal Bin Widths
Using different interval sizes (e.g., \(0\text{--}10\), \(10\text{--}30\), \(30\text{--}35\)). All bins must be the same width.
✕ Wrong Quartiles
Forgetting to order data first, or including the median in both halves when finding \(Q_1\) and \(Q_3\).
✕ Dot Plot for Large Data
Using a dot plot for \(100+\) data points. The dots become unreadable: use a histogram instead.
✕ Missing Labels
Forgetting the title, axis labels, or units. Without labels, a display cannot be interpreted.

Summary at a Glance

Feature Dot Plot Histogram Box Plot
Data type Numerical Numerical Numerical
Best for Small sets, exact values Large sets, distribution shape Comparing groups, spread
Key feature Every value visible Bars show frequency Five-number summary
Identifies outliers? Visually Hard to see Yes (\( 1.5 \times \text{IQR} \))
Shows mean? No No No (shows median)

Question 1 of 5

A teacher records the number of books read by \(15\) students over summer break. Which data display would best show every individual value?

Question 2 of 5

In a histogram, why must the bars touch each other with no gaps?

Question 3 of 5

Given the ordered data set: \(4, 7, 9, 12, 15, 18, 21, 24, 30\), what is the interquartile range (IQR)?

Question 4 of 5

A box plot shows \( Q_1 = 20 \), \( Q_3 = 40 \), and \( \text{IQR} = 20 \). Using the \( 1.5 \times \text{IQR} \) rule, which value would be considered an outlier?

Question 5 of 5

You want to compare the distribution of test scores between three different classes. Which data display is the best choice?