Dot plots

You ask $30$ friends how many cups of water they drank yesterday. You could write a list of $30$ numbers , boring and hard to read. Or you could draw a number line and place one dot above each value. Suddenly the shape of the pile of dots reveals patterns: are most people drinking $4-6$ cups? Is anyone drinking $10+$? A dot plot is one of the simplest, friendliest data pictures, and you should always reach for it first when working with a small list of numbers.

Concept

A dot plot is a 1-dimensional picture of data. The horizontal axis is a number line. For each value in the data, you place one dot above that number on the line. When several values are equal, the dots stack on top of one another.

Reading a dot plot.

• The mode (most common value) is the position with the tallest stack.
• The range is from the leftmost dot to the rightmost.
• The mean can be found by computing $\bar{x} = \dfrac{\sum x_i}{n}$, but more usefully you can eyeball its position as the balance point of the dots.
• The median is the dot at the middle position when you count from either end.
• Outliers appear as isolated dots far from the main cluster.

Drawing a dot plot.

1. Find the smallest and largest values; draw a number line covering that range.
2. Mark equal divisions (often integers).
3. For each value in the data, place a dot above the matching tick.
4. Title the plot and label the axis.

What dot plots are good for.

• Small to medium datasets (say, up to $50$ values).
• Whole-number or simply-spaced data.
• Quickly spotting the typical value, spread, and outliers.

When to use something else.

• Long lists with rare repeats , dots stretch out flat and tell you little.
• Continuous data with many decimal places , values rarely repeat exactly.
• Comparing across time , use a line graph instead.

Shapes of dot plots.

• A single peak with symmetric drop-off suggests a typical "middle" value with random spread.
• A flat plot suggests all values are roughly equally likely.
• A double peak suggests two different groups mixed together.
• A long tail to the right suggests a few unusually large values.

Worked examples

Example 1. Data: $4, 6, 7, 7, 8, 8, 8, 9, 10$. Describe the dot plot.

• Range $4$ to $10$. Tallest stack at $8$ (three dots). Mode $= 8$.
• Sorted, so median is the $5$th value $= 8$. Mean $= 67/9 \approx 7.44$.
• The plot peaks around $8$, with single dots tailing off on both sides.

Example 2. A dot plot has dots at $4, 8, 8, 8, 9, ?$ and mean $9$. Find the missing dot.

• Sum needed $= 6 \times 9 = 54$. Known sum $= 4+8+8+8+9 = 37$. Missing $= 17$. The dot sits far right at $17$.

Example 3. A dot plot shows the lengths (in minutes) of $8$ songs: $5, 5, 5, 5, 5, 5, 5, 5.5$. Estimate the mean.

• Sum $= 35 + 5.5 = 40.5$. Mean $= 40.5/8 \approx 5.06$ minutes , the plot is very tightly packed.

Example 4. Heights of $10$ students (in cm): $145, 148, 150, 150, 152, 152, 152, 155, 158, 160$. Find mode, median, mean.

• Mode $= 152$ (appears thrice). Median (between $5$th and $6$th values) $= (152+152)/2 = 152$. Mean $= 1522/10 = 152.2$.
• The three measures are all near $152$ , the data is fairly symmetric.

Try it yourself

1. Draw the dot plot for: $1, 2, 2, 3, 3, 3, 4, 4, 5$. Find mode and median.
2. Find the mean of the data above and mark its position on the dot plot.
3. Twenty students score (out of $10$): five score $7$, eight score $8$, four score $9$, three score $10$. Sketch the dot plot. Find the mean.
4. A dot plot has dots at $3, 4, 4, 5, 5, 6, 6, 9, 12, 15$. Which is greater , the mean or the median?
5. A dot plot has its tallest stack at $0$ (four students cycled $0$ times), then dots at $1, 1, 2, 2, 3, 3, 4, 4, 5, 6$. How many students rode their cycle at least once?
6. Sketch the dot plot of $5, 7, 7, 7, 9, 9, 11, 11, 11, 13$. Is the shape symmetric?
7. Find the mode of $2, 3, 3, 4, 5, 5, 5, 6, 7, 8$.
8. Which is easier to read off from a dot plot , the mean, median, or mode? Why?

Activity

Family size in the class. Each student writes the number of people in their household on a small card. On a single number line on the board, place one chalk-dot per card at the right position. The dot plot of the whole class will appear in a couple of minutes , see what the mode is. Is anyone an outlier? Compute the mean and median and mark both on the same line.