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Dot Plots, Histograms, and Box Plots

Read one-variable data on a number line and find counts, percents, median, quartiles, range, and IQR from dot plots, histograms, and box plots.

Practice this skill 30 min Dot plots, histograms and box plots Leer en español
Key idea

What you'll learn

When you collect one kind of number — test scores, temperatures, commute times — you need a way to see the whole set at once. Three graphs do this using a number line: the dot plot, the histogram, and the box plot.

In this lesson you will learn to:

  • read a dot plot and count values, find the mode, median, mean, and range from it,
  • read a histogram, which groups data into intervals, and answer "how many" and "what percent" questions,
  • read a box plot and its five-number summary, and find the range and the interquartile range (IQR),
  • compare two data sets using their center and their spread, and
  • describe a data set's shape (symmetric, skewed) and spot outliers.

Why it matters on the GED: The GED regularly shows one of these graphs and asks you to read a value, find a measure of center or spread, or compare two groups. The trap answers are predictable: using the tallest bar as the median, confusing range with IQR, or thinking a longer part of a box holds more data. In real life, you see these displays in health reports (blood pressure ranges), pay surveys, and school reports.

Words to know

  • Data set — a list of numbers about one thing (the test scores of a class).
  • Dot plot (line plot) — a number line with one dot for each data value, stacked when values repeat. (Three dots above 4 means the value 4 appears three times.)
  • Frequency — how many times a value or interval occurs.
  • Histogram — a bar graph for numbers grouped into equal intervals (bins), such as 10–under 20 minutes. The bars touch.
  • Interval (bin) — a range of values, such as 20 to under 30.
  • Box plot (box-and-whisker plot) — a graph of the five-number summary.
  • Five-number summary — minimum, first quartile (Q_1), median, third quartile (Q_3), maximum.
  • Median — the middle value when the data are in order (for 1, 3, 8 it is 3).
  • Quartiles — values that split ordered data into four equal parts. Q_1 is the median of the lower half, Q_3 the median of the upper half.
  • Range — maximum − minimum (for 52 to 96: 96 - 52 = 44).
  • Interquartile range (IQR) — Q_3 - Q_1, the spread of the middle half (for 70 and 84: 14).
  • Mode — the value that occurs most often (the tallest stack of dots).
  • Outlier — a value far away from the rest (13 books when everyone else read 0 to 6).
  • Skewed — lopsided: a long tail stretching to one side.

All Math words in the glossary

Dot plots

A dot plot puts every data value on a number line. Each dot is one value, and dots stack when a value repeats.

Dot plot of the number of books read last month by 20 students. Each dot is one student. 0 books: 2 dots; 1: 3 dots; 2: 5 dots; 3: 4 dots; 4: 3 dots; 5: 2 dots; 6: 1 dot.0123456Books read last month

Reading it. The number under a column is the value; the number of dots is its frequency. Above, 5 students read 2 books.

Counting. Add dots over the values you need. Pay attention to boundary words:

PhraseIncludes the boundary?"___ 3 books" means
more than / greater thanno4, 5, 6
at least / 3 or moreyes3, 4, 5, 6
fewer than / less thanno0, 1, 2
at most / no more thanyes0, 1, 2, 3

Example: At least 3 books: 4 + 3 + 2 + 1 = 10 students.

Mode = tallest column = 2 books. Range = 6 - 0 = 6 books.

Median. The dots are already in order. Count to the middle position.

  • Odd number of values n: the median is value number \tfrac{n+1}{2}.
  • Even n: average values number \tfrac{n}{2} and \tfrac{n}{2} + 1.

Example: n = 20, so average the 10th and 11th dots. The 10th is the last dot over 2; the 11th is the first dot over 3. Median = \tfrac{2 + 3}{2} = 2.5.

The books dot plot (20 students) with a running count under each column: 2, 5, 10, 14, 17, 19, 20. The 10th dot is the top dot over 2 and the 11th dot is the bottom dot over 3; both are highlighted in red. The median is the average of 2 and 3, which is 2.5, marked with a red arrow on the number line.0123456251014171920Running count:10th11thmedian = (2 + 3) ÷ 2 = 2.5Books read last month (each dot = 1 student)

Mean. Multiply each value by its number of dots, add, and divide by the total number of dots: \tfrac{0(2) + 1(3) + 2(5) + 3(4) + 4(3) + 5(2) + 6(1)}{20} = \tfrac{53}{20} = 2.65.

Histograms

A histogram groups numbers into equal intervals and draws a bar for each. The height of a bar is the frequency — how many values fall in that interval.

Histogram of commute times for 50 workers. Intervals include the left edge but not the right edge. 0 to under 10 minutes: 5 workers; 10 to under 20: 17; 20 to under 30: 12; 30 to under 40: 8; 40 to under 50: 5; 50 to under 60: 3. Vertical axis: number of workers, 0 to 18 in steps of 2.0246810121416180102030405060Commute time (minutes)Number of workers

Reading it. The bar from 10 to 20 reaches 17, so 17 workers commute from 10 to under 20 minutes. You do not know any single worker's exact time.

Total count = sum of all bar heights: 5 + 17 + 12 + 8 + 5 + 3 = 50.

Counting over several intervals. Example: At least 30 minutes: 8 + 5 + 3 = 16 workers. Under 20 minutes: 5 + 17 = 22 workers, which is \tfrac{22}{50} = 44\%.

Which interval holds the median? Find the middle position, then keep a running total from the left. For 50 values, the median is between the 25th and 26th. Running totals: 5, 22, 34. The 25th and 26th values are in the 20–30 interval.

Modal interval = the tallest bar (10–20 here). It is not the same as the median interval.

Shape. A histogram (or dot plot) shows the shape of the data:

  • Symmetric — the left and right sides look like mirror images; mean ≈ median.
  • Skewed right — a long tail to the right (a few very long commutes); the mean is pulled above the median.
  • Skewed left — a long tail to the left; the mean is pulled below the median.
Three histogram shapes. Symmetric: bars rise to a peak in the middle and fall the same way, so mean and median are about equal. Skewed right: tall bars on the left and a long tail to the right, so the mean is pulled above (to the right of) the median. Skewed left: a long tail to the left, so the mean is pulled below (to the left of) the median.Symmetricmean ≈ medianSkewed rightmean > mediantail →Skewed leftmean < median← tailmeanmedian

The commute histogram is skewed right: most people have short or medium commutes, and a few have long ones.

Histogram vs. bar graph. A bar graph compares categories (car, bus, train) with gaps between bars. A histogram shows number intervals in order, and the bars touch because the intervals share edges.

Box plots and the five-number summary

A box plot squeezes a whole data set into five numbers.

Labeled box plot of test scores: minimum 52, first quartile Q1 70, median 76, third quartile Q3 84, maximum 96. The box runs from Q1 to Q3 and the whiskers reach the minimum and maximum.50556065707580859095100Test scoreMinimum = 52Q1 = 70Median = 76Q3 = 84Maximum = 96
Part of the plotNumberValue here
left whisker endminimum52
left edge of boxQ_170
line inside boxmedian76
right edge of boxQ_384
right whisker endmaximum96

The key fact: the four sections — left whisker, left half of box, right half of box, right whisker — each hold about 25% of the data. So:

  • about 50\% of scores are between Q_1 and Q_3 (in the box),
  • about 25\% are above Q_3, and about 75\% are below Q_3.
Box plot of test scores: minimum 52, Q1 70, median 76, Q3 84, maximum 96. Brackets above show that each of the four sections (52 to 70, 70 to 76, 76 to 84, 84 to 96) holds about 25% of the scores, even though the sections have different lengths. Below, the box from Q1 to Q3 is the IQR, 84 minus 70 = 14, and the range is 96 minus 52 = 44.5055606570758085909510025%25%25%25%Min 52Q₁ 70Med 76Q₃ 84Max 96IQR = 84 − 70 = 14

Length shows spread, not count. The right half of the box (76 to 84) is wider than the left half (70 to 76), but each holds about a quarter of the scores. Wider just means those scores are more spread out.

Measures of spread:

  • Range = \text{max} - \text{min} = 96 - 52 = 44.
  • IQR = Q_3 - Q_1 = 84 - 70 = 14.

Building a five-number summary from a list. Data: 3, 5, 6, 8, 9, 11, 12, 15 (8 values, in order).

  1. Min = 3, max = 15.
  2. Median = \tfrac{8 + 9}{2} = 8.5.
  3. Lower half: 3, 5, 6, 8 → Q_1 = \tfrac{5 + 6}{2} = 5.5. Upper half: 9, 11, 12, 15 → Q_3 = \tfrac{11 + 12}{2} = 11.5.
  4. IQR = 11.5 - 5.5 = 6.
The 8 values 3, 5, 6, 8, 9, 11, 12, 15 as dots on a number line. The median 8.5 falls between 8 and 9. The lower half 3, 5, 6, 8 has Q1 = 5.5 (between 5 and 6); the upper half 9, 11, 12, 15 has Q3 = 11.5 (between 11 and 12). The box plot below shows min 3, Q1 5.5, median 8.5, Q3 11.5, max 15, with IQR 11.5 minus 5.5 = 6.2345678910111213141516Q₁ 5.5Median 8.5Q₃ 11.5lower halfupper halfMin 3Max 15IQR = 11.5 − 5.5 = 6

(With an odd number of values, leave the median out of both halves.)

Comparing two box plots. Compare center using the medians and spread using the IQRs (or ranges). One group can have a higher center but less spread.

Outliers. A common rule calls a value an outlier if it is more than 1.5 \times \text{IQR} below Q_1 or above Q_3. Here 1.5 \times 14 = 21; outliers would be below 70 - 21 = 49 or above 84 + 21 = 105. None of these scores are outliers.

Formula

Formulas (what is and isn't on the sheet)

\text{Range} = \text{maximum} - \text{minimum} \text{IQR} = Q_3 - Q_1 \text{Median position} = \frac{n + 1}{2} \text{Mean from a dot plot or frequency table} = \frac{\sum (\text{value} \times \text{frequency})}{\sum \text{frequency}}
  • n — the number of data values; if \tfrac{n+1}{2} is a half (like 10.5), average the two values on either side.
  • Q_1, Q_3 — the first and third quartiles (edges of the box).
  • \sum — "add up all of these."

The GED formula sheet only defines the median (the middle value of an ordered set, or the mean of the two middle values) and the mean. Range, IQR, quartiles and the median-position shortcut are not on the sheet — memorize them.

Worked example

Example 1 (easy): Read a dot plot

Using the books dot plot, how many students read at most 1 book?

  1. "At most 1" includes 0 and 1.
  2. Dots over 0: 2. Dots over 1: 3.
  3. Add: 2 + 3 = 5.
Example 1. The books dot plot with the columns over 0 (2 dots) and 1 (3 dots) shaded red: "at most 1 book" includes 0 and 1, so 2 + 3 = 5 students. The other 15 dots, over 2 through 6, are blue.0123456at most 1: 2 + 3 = 52 or more: 15Books read

Answer: 5 students.

Check: The remaining students read 2 or more: 5 + 4 + 3 + 2 + 1 = 15, and 5 + 15 = 20 ✓

Worked example

Example 2 (medium): Box plot — real world

A box plot of a store's daily customers has min 40, Q_1 55, median 62, Q_3 75, max 110. Over 120 days, on about how many days did the store have more than 75 customers?

  1. 75 is Q_3. About 25\% of the data lie above Q_3.
  2. 25\% of 120 days: 0.25 \times 120 = 30.
Example 2. Box plot of daily customers over 120 days: min 40, Q1 55, median 62, Q3 75, max 110. The right whisker, from 75 to 110, is shaded red: about 25% of the days, 0.25 × 120 = 30 days, had more than 75 customers.405060708090100110Min40Q₁55Med62Q₃75Max110about 25% → 0.25 × 120 = 30 daysCustomers per day

Answer: about 30 days.

Check: About 75\% of days, or 0.75 \times 120 = 90 days, are at or below 75, and 90 + 30 = 120 ✓

Worked example

Example 3 (medium): Histogram percent

Using the commute histogram (50 workers), what percent of workers commute from 20 to under 40 minutes?

  1. Bars for 20–30 and 30–40: 12 + 8 = 20 workers.
  2. Percent: \dfrac{20}{50} = 0.40 = 40\%.
Example 3. The commute histogram for 50 workers with the bars for 20 to 30 minutes (12) and 30 to 40 minutes (8) shaded red: 12 + 8 = 20 workers, and 20 ÷ 50 = 40%. Other bars: 0 to 10 has 5, 10 to 20 has 17, 40 to 50 has 5, 50 to 60 has 3.0246810121416180102030405060Commute time (minutes)Number of workers5171285312 + 8 = 20 of 50 = 40%

Answer: 40%.

Check: 40\% of 50 is 0.4 \times 50 = 20 ✓

Worked example

Example 4 (GED-hard): Five-number summary from data — word problem

A nurse records the ages of 9 patients: 34, 61, 28, 45, 52, 39, 70, 47, 55. Find the median and the IQR.

  1. Put in order: 28, 34, 39, 45, 47, 52, 55, 61, 70.
  2. n = 9; median is position \tfrac{9+1}{2} = 5 → 47.
  3. Lower half (leave out 47): 28, 34, 39, 45 → Q_1 = \tfrac{34 + 39}{2} = 36.5.
  4. Upper half: 52, 55, 61, 70 → Q_3 = \tfrac{55 + 61}{2} = 58.
  5. IQR = 58 - 36.5 = 21.5.
Example 4. The 9 patient ages in order as dots on a number line: 28, 34, 39, 45, 47, 52, 55, 61, 70. The median is the 5th value, 47. Q1 = 36.5 is halfway between 34 and 39; Q3 = 58 is halfway between 55 and 61. The box plot below shows min 28, Q1 36.5, median 47, Q3 58, max 70, and IQR 58 minus 36.5 = 21.5.2530354045505560657075Q₁ 36.5Median 47Q₃ 58Min 28Max 70IQR = 58 − 36.5 = 21.5

Answer: median 47 years, IQR 21.5 years.

Check: 4 ages lie below 47 and 4 above ✓; Q_1 and Q_3 each split their half into 2 and 2 ✓

Worked example

Example 5 (GED-hard): Work backwards — missing dots

A dot plot of quiz scores shows: score 6 (2 dots), 7 (3 dots), 8 (5 dots), 9 (4 dots), and 10 (a smudged column — the number of dots can't be read). The teacher says the mean score is 8.45. How many students scored 10?

  1. Let k = the number of dots above 10. Number of students: 2 + 3 + 5 + 4 + k = 14 + k.
  2. Total of the scores: 6(2) + 7(3) + 8(5) + 9(4) + 10k = 12 + 21 + 40 + 36 + 10k = 109 + 10k.
  3. Set the mean equal to 8.45 and solve: \frac{109 + 10k}{14 + k} = 8.45 109 + 10k = 8.45(14 + k) = 118.3 + 8.45k 1.55k = 9.3 \quad\Rightarrow\quad k = 6
Example 5. Dot plot of quiz scores: 6 has 2 dots, 7 has 3, 8 has 5, 9 has 4, and the smudged column over 10 holds k dots, found to be 6 (drawn dashed in red). With 20 students the total is 169, and the mean 8.45 is the balance point, marked with a triangle under the number line.678910k = 6(smudged)mean 8.45 (balance point)(109 + 10k) ÷ (14 + k) = 8.45Quiz score

Answer: 6 students scored 10.

Check: 14 + 6 = 20 students; total = 109 + 60 = 169; 169 \div 20 = 8.45 ✓

Calculator tip

Using the TI-30XS MultiView

Reading these graphs is done by eye, so most of the work is counting. Use the calculator for the arithmetic afterwards:

  • Mean from a dot plot: type the products in one line: 0 × 2 + 1 × 3 + 2 × 5 + … enter to get the total (53), then ÷ 20 enter → 2.65. Put the whole sum in parentheses first if you want one step: ( … ) ÷ 20.
  • Percent of a histogram: 22 ÷ 50 enter → 0.44; that is 44\%.
  • If a result appears as a fraction, press toggle to see the decimal.

When not to use it: Finding a median or a quartile is about position, not arithmetic — count dots or list the data in order on your erasable note board. The calculator cannot count dots for you.

Common mistake

Common mistakes

  • Using the tallest bar or column as the median. Wrong: "the 10–20 bar is tallest, so the median is there." Right: the tallest bar is the mode; find the median by a running total to the middle position.
  • Confusing range and IQR. Wrong: IQR = 96 - 52 = 44. Right: IQR uses the box: 84 - 70 = 14; range uses the whiskers.
  • Thinking a longer section holds more data. Wrong: "the right half of the box is wider, so more students are there." Right: each of the four sections holds about 25\%; wider means more spread out.
  • Reading the mean from a box plot. Wrong: "the mean is 76." Right: the line in the box is the median; a box plot doesn't show the mean.
  • Dividing by the number of columns. Wrong: mean = 53 \div 7. Right: divide by the number of dots (data values), 20.
  • Including or excluding the boundary wrongly. "More than 3" does not include 3; "at least 3" does.
Test strategy

GED test strategy

  • Identify the graph first. Dot plot → count dots. Histogram → add bar heights over intervals. Box plot → read the five numbers. Write them down.
  • Read the question twice and circle the measure it asks for: median, mean, mode, range, or IQR. The wrong-answer choices are usually the other measures.
  • Use the 25% rule for box plots: any "what percent" or "about how many" question about a box plot is usually answered with 25%, 50%, or 75%.
  • Check your total. The sum of dots or bar heights must equal the number of data values in the question.
  • Eliminate impossible answers. A mean or median can't be smaller than the minimum or bigger than the maximum.
  • Watch the scale on the number line — it may count by 2s, 5s, or 10s.
  • Time: most of these take under 90 seconds. If a count gets long, write a running total instead of recounting.

Check yourself

Dot plots, histograms and box plots

The dot plot shows how many books each of 20 students read last month. Each dot stands for one student.

Dot plot of the number of books read last month by 20 students. Each dot is one student. 0 books: 2 dots; 1: 3 dots; 2: 5 dots; 3: 4 dots; 4: 3 dots; 5: 2 dots; 6: 1 dot.0123456Books read last month

How many students read more than 3 books?

Answer choices
Hint

"More than 3" does not include 3 itself. Count the dots in each column to the right of 3.

Check yourself

Dot plots, histograms and box plotsFill in

The dot plot shows how many books each of 20 students read last month. Each dot stands for one student.

Dot plot of the number of books read last month by 20 students. Each dot is one student. 0 books: 2 dots; 1: 3 dots; 2: 5 dots; 3: 4 dots; 4: 3 dots; 5: 2 dots; 6: 1 dot.0123456Books read last month

Look only at the students who read at least 1 book. What is the median number of books read by these students?

books

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Leave out the students who read 0 books. Count how many students are left, then find the middle position.

Summary

  • A dot plot shows every value; count dots for frequencies and count to the middle position for the median.
  • A histogram groups values into intervals; bar height = how many values are in the interval; add heights to count.
  • A box plot shows min, Q_1, median, Q_3, max; each of the four sections holds about 25\% of the data.
  • Range = max - min; IQR = Q_3 - Q_1 (the middle half).
  • Compare groups by center (median) and spread (IQR or range); skew and outliers pull the mean toward the tail.
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