Building a Stemplot Without Losing Values

Why a Stemplot Preserves the Data

A graph can make a collection of measurements easier to scan, but some displays combine values into groups. A stemplot organizes quantitative observations while retaining each individual value: the first part of each value forms a stem, and the remaining part forms a leaf. For two-digit whole-number temperatures, the tens digit is the stem and the ones digit is the leaf.

For example, the temperature \(54^\circ\text{F}\) is represented by stem 5 and leaf 4. The bar between the stem and leaf is a divider, not a decimal point. Once a reader knows the key, the original temperature can be reconstructed. The leaves are also arranged in order, making the display orderly without changing or discarding the measurements.

A stemplot is useful when it is important to keep the individual measurements visible, rather than displaying only totals in intervals. The construction in this tutorial is for two-digit whole-number temperatures. Keep the measurement units and precision consistent: a list that mixes Fahrenheit and Celsius, or whole degrees and tenths of a degree, cannot be represented unambiguously by the same stem-and-leaf rule without first clarifying how the values are recorded.

A Consistent Construction Method

Start with the raw observations and decide what each digit will mean. For two-digit temperatures, use the tens digit as the stem and the ones digit as the leaf. Then list the stems in increasing order and place each observation's leaf on the row for its stem. Within every row, sort the leaves from smallest to largest. Do not remove repeats: two equal observations produce two identical leaves.

The stems ordinarily run from the smallest observed stem to the largest. If no observation has a stem between those endpoints, show that stem with no leaves. Keeping the row makes the numerical scale continuous and makes the absence of observations in that row explicit. Finish with a key, then count the leaves and compare that count with the number of raw observations.

Repeats Are Separate Observations

A stemplot represents observations, not just different values. If the same temperature occurs several times, each occurrence needs its own leaf. For instance, three readings of \(72^\circ\text{C}\) contribute three 2s on the stem-7 row. Removing duplicate leaves would change the data: it would conceal how many times that temperature was recorded.

Why the Key and Place Values Matter

A stemplot is only readable if the key establishes how to interpret its marks. The line \(5|4\) could mean 54 degrees, 5.4 degrees, or something else if no key is given. A key both removes that ambiguity and names the units. With the rule for two-digit whole numbers, the stem gives the tens and the leaf gives the ones.

The reconstruction rule can be written precisely. If \(s\) is the stem and \(l\) is the leaf, the recorded two-digit value is \(10s+l\). For example, stem 6 and leaf 3 give \(10(6)+3=63\). This follows from place value: six tens and three ones make 63. Therefore, when every observation is assigned its actual tens and ones digits and a key states the rule, the display retains the original values.

Sorting leaves changes only their visual order, not their values or their number. A useful final check has two parts: each leaf must match the correct stem according to the raw measurement, and the total number of leaves must equal the total number of observations. The key makes it possible to verify that a leaf such as 3 on stem 6 means 63, rather than an adjacent value.

Common Mistakes and a Final Check

The most common construction errors are small but consequential. A misplaced leaf changes a value; a missing repeated leaf changes the frequency of a value; and a missing or vague key can make the whole display impossible to interpret. Before presenting a stemplot, compare the rows with the raw data rather than relying only on how the finished display looks.

• Dropping repeats. Wrong reasoning: “The value 72 is already shown, so another 2 is unnecessary.” Correction: every observation contributes a leaf, so three readings of 72 require three 2s on stem 7.
• Sorting the data but not the leaves. Wrong reasoning: “The values are all present, so their order on the row does not matter.” Correction: leaves should be in increasing order within each stem so the display follows a consistent, readable format.
• Leaving out intermediate stems. Wrong reasoning: “A row with no data has no purpose.” Correction: include empty stems between the smallest and largest stems to show the scale without skipping tens groups.
• Omitting the key or units. Wrong reasoning: “Everyone knows what the divider means.” Correction: explicitly state an example reconstruction and the units, such as \(5|4=54^\circ\text{F}\).

A finished stemplot should let a reader recover each original measurement, including repeated measurements. If a reader cannot tell what \(6|3\) means, or if the number of leaves does not match the number of observations, the display needs correction before it is used.