Diagrama de tallo y hojas: explicación y ejemplos

The definition of the stem and leaf plot is:

**“The stem and leaf plot is a chart used to represent numerical data to show its distribution”**

In this topic, we will discuss the line graph from the following aspects:

- What is a stem and leaf plot?
- How to read a stem and leaf plot?
- How to make a stem and leaf plot?
- Types of stem and leaf plots
- Preguntas practicas
- respuestas

## What is a stem and leaf plot?

The stem and leaf plot is a plot used to represent numerical data by showing its distribution.

Each numerical data value is split into a stem (the first digit or digits) and a leaf.

The stem is the first digit or digits, while the leaf is the last digit.

The stem and leaf plot is used when your data is not too large (about 15-150 data points).

The stem and leaf plot is drawn in a table with two columns.

The stems are listed down in the left column. Each stem is listed, even if some stems have no leaves.

The leaves are listed in increasing order in a row to the right of each corresponding stem.** **

**Ejemplo**, the following is the age in years for 15 persons from a certain survey.

70 56 37 69 70 40 66 53 43 70 54 42

If we plot this data as a stem and leaf plot, we will get

Key: 3|7 means 37 years

Here, the stem unit represents tens and the leaf unit represents single values.

The 3 stem can represent any number from 30 to 39.

Stem 3, leaf 7 means 37.

Stem 4, leaf 0 means 40.

Stem 5, leaf 3 means 53.

Stem 7, leaf 0 means 70.

From this stem plot, we can conclude that:

- The minimum age is 37 years and the maximum age is 70 years.
- The most frequent age (or the mode) in this data is 70 years because it occurs 3 times. There is no other value that occurs more than that.

## How to read a stem and leaf plot?

Veamos un ejemplo:

The following is a stem and leaf plot of the heights in cm of 30 participants

Key: 14|7 means 147 cm.

- We look at the key, the stem represents tens and the leaf represents single values.
- Look at the first row to get the minimum of our data. The minimum = 147 cm.
- Look at the last row to get the maximum of our data. The maximum = 180 cm.
- Look at the most frequent value in each row to get the most frequent value in our data or the mode.

There are 4 zeros beside 16 so the mode in this data is 160 cm because it is repeated 4 times. There is no other value that is repeated more than that.

- Look at the crowded rows to see where the main cluster of data.

The data are clustered at 15s and 16s or from 150-169.

150 is the minimum value for row 15 to represent and 169 is the maximum value that row 16 can represent.

15 has 11 numbers in its row and 16 has 13 numbers in its row.

Lower and larger values are at low frequency or rare in our data.

**Otro ejemplo**, the following is a stem and leaf plot of 30 wind measurements in miles per hour (mph) in New York City.

Key: 5|7 = 5.7.

- We look at the key, the stem represents the single values and the leaf represents the decimal values.
- Look at the first row to get the minimum of our data. The minimum = 5.7 mph.
- Look at the last row to get the maximum of our data. The maximum = 20.1 mph.
- Look at the most frequent value in each row to get the most frequent value in our data or the mode.

There are 4 sevens beside 9 so the mode in this data is 9.7 because it is repeated 4 times. There is no other value that is repeated more than that.

- Look at the crowded rows to see where the main cluster of data.

The data are clustered at 9s,11s, and 14s or from 9.0 to 14.9.

9.0 is the minimum value for row 9 to represent, and 14.9 is the maximum value for row 14 to represent.

Lower and larger values are at low frequency or rare in our data.

## How to make a stem and leaf plot?

We will follow some steps through an example:

The following are the body mass indices (BMI)of 10 individuals

25.0, 25.2, 24.2, 31.5, 17.4, 29.4, 19.2, 20.7, 24.2, 29.7

Let’s make a stem and leaf plot of this data

- The data are sorted in ascending order.

17.4, 19.2, 20.7, 24.2, 24.2, 25.0, 25.2, 29.4, 29.7, 31.5

- Find the largest and smallest number in the data.

The smallest value is 17.4 and the largest value is 31.5

- Determined what the stems will represent and what the leaves will represent.

Each stem can consist of any number of digits, but each leaf can have only the single last digit.

If the range of values is too great, the numbers can be rounded up to limit the number of stems.

In this example, the leaf represents the decimal place and the stem will represent the rest of the number (ones and tens place).

- The minimum of our data is 17.4 (which contains 17 in the ones place) and the maximum is 31.5 (which contains 31 in the ones place) so our stems must go from 17 to 31. It will contain about 14 rows.
- The stem and leaf plot is drawn with two columns. The stems are listed down in the left column (from 17 to 31).

- Separate each data value into a stem (of ones and tens) and a leaf (of decimal points).

For the data value, 17.4, the stem is 17, and 4 is the leaf. Write 4 in the row of 17 stem.

The next data value, 19.2, the stem is 19, and 2 is the leaf. Write 2 in the row of 19 stem.

- The leaves are listed in increasing order in a row to the right of each stem in the right column.

Continue till all data values are listed in the stem and leaf plot. Write a key at the bottom of the table.

Key: 17|4 = 17.4

There are some stems that are empty, 18,21,22,23,26,27,28, and 30 as they have no corresponding values.

**Example of rounding used to limit the number of stems**

The following is the balance account of 10 clients from a certain bank

143, 29, 2, 506, 1, 231, 447, 2, 121, 593

Let’s make a stem and leaf plot of this data

- The data are sorted in ascending order.

1, 2, 2, 29, 121, 143, 231, 447, 506, 593

- Find the largest and smallest number in the data.

The smallest value is 1 and the largest value is 593.

- Determined what the stems will represent and what the leaves will represent.

In this example, we can set the leaves to represent ones and the stem to represent the rest of the number (tens and hundreds).

- The data minimum is 1 (which contains 0 in the tens place) and the maximum is 593 (which contains 59 in the tens place) so our stems must go from 0 to 59. This means that it will contain 60 rows.
- The stem and leaf plot is drawn with two columns. The stems are listed down in the left column (from 0 to 59).

- Separate each data value into a stem (of tens) and a leaf (of ones).

For the data value, 1, the stem is 0 as it has no tens, and 1 is the leaf. Write 1 in the row of 0 stem.

The next data value, 2, the stem is 0, and 2 is the leaf. Write 2 in the row of 0 stem.

The next data value, 2, the stem is 0, and 2 is the leaf. Write another 2 in the row of 0 stem.

The next data value, 29, the stem is 2, and 9 is the leaf. Write 9 in the row of 2 stem.

Continue till all data values are listed in the stem and leaf plot. Write a key at the bottom of the table.

Key: 59|3 = 593

- The table is very long and very hard to read. So we use rounding to the nearest tens, so the stems will represent hundreds and the leaves tens. This will reduce the number of stems.

- After rounding, the data minimum is 0 (which contains 0 in the hundreds place) and the maximum is 590 (which contains 5 in the hundreds place) so our stems must go from 0 to 5. This means that it will contain only 6 rows.
- The stem and leaf plot is drawn with two columns. The stems are listed down in the left column (from 0 to 5).

- Separate each (rounded) data value into a stem (of hundreds) and a leaf (of tens).

For the data value, 0, the stem is 0 as it has no hundreds, and 0 is the leaf also. Write 0 in the row of 0 stem.

For the next data value, 0, write another 0 in the row of 0 stem.

For the next data value, 0, write another 0 in the row of 0 stem.

The next data value, 30, the stem is 0 as it has no hundreds, and 3 is the leaf or tens. Write 3 in the row of 0 stem.

The next data value, 120, the stem is 1 as it has 1 as hundred and 2 is the leaf or tens. Write 2 in the row of 1 stem.

Continue till all data values are listed in the stem and leaf plot. Write a key at the bottom of the table.

And the stem and leaf plot will be

Key: 0|3 = 30, 1|2 = 120

- The 0 stem and 0 leaf mean that original values are less than 5, so rounded to 0.
- The 0 stem includes the rounded values from 0-90.
- 1 stem includes rounded values from 100-190.
- 2 stem includes rounded values from 200-290, and so on.

**Example of rounding with negative values**

The following is the balance of 10 clients from a certain bank

-7, -3, 506,0, 2586,49, 104,529, -171, -364

Create a stem and leaf plot for this data

- The data are sorted in ascending order.

-364, -171, -7, -3, 0, 49, 104, 506, 529, 2586

- Find the largest and smallest number in the data.

The smallest value is -364 and the largest value is 2586.

- Determined what the stems will represent and what the leaves will represent.

In this example, we can set the leaves to represent ones and the stem to represent the rest of the number (tens, hundreds, and thousands).

- The data minimum is -364 (which has -36 in the tens place) and the maximum is 2586 (which has 258 in the tens place) so our stems must go from -36 to 258. This means that it will contain about 295 rows. This is an incredibly large table and will be difficult to read.
- We use rounding to the nearest tens, so the stems will represent hundreds and the leaves tens. This will reduce the number of stems.

Note that values from -4 to -1 are rounded to -0.

Values from 1 to 4 are rounded to 0.

- After rounding, the data minimum is -360 (which contains -3 in the hundreds place) and the maximum is 2590 (which contains 25 in the hundreds place) so our stems (which now are representing hundreds) must go from -3 to 25. This means that it will contain about 28 rows.
- The stem and leaf plot is drawn with two columns. The stems are listed down in the left column (from -3 to 25).

- Separate each rounded data value into a stem (of hundreds) and a leaf (of tens).

The first (rounded) data value, -360, the stem is -3 as it has -3 in the hundreds place and 6 is the leaf as it has 6 in the tens place. Write 6 in the row of -3 stem.

The next data value, -170, the stem is -1, and 7 is the leaf or tens. Write 7 in the row of -1 stem.

The next data value, -10, the stem is -0 (as it has no hundred value and the negative sign of -0 to indicate that it is a negative value) and 1 is the leaf or tens. Write 1 in the row of -0 stem.

The next data value, -0, the stem is -0 and 0 is the leaf. Write 0 in the row of -0 stem.

Key: 25|9 = 2590

- The -3 stem includes the (rounded) values from -390 to -300.
- The -2 stem includes the values from -290 to -200.
- The -1 stem includes the values from -190 to -100.
- The -0 stem includes the values from -90 to -0.
- The 0 stem includes the values from 0 to 90.
- The 1 stem includes the values from 100 to 190.
- The 2 stem includes the values from 200 to 290, and so on.

- We may see that our stem and leaf plot is still large. We use rounding to the nearest hundreds, so the stems will represent thousands and leaves hundreds. This will reduce the number of stems further.

In that case, values from -49 to -1 are rounded to -0, and values from 1 to 49 are rounded to 0.

values from -50 to -149 are rounded to -1 (meaning -100) and values from 50 to 149 are rounded to 1 (meaning 100).

- The data minimum is -400 (which contains 0 in the thousands place) and the maximum is 2600 (which contains 2 in the thousands place) so our stems (which now are representing thousands) must go from -0 to 2. This means that it will contain only 4 rows.
- The stem and leaf plot is drawn with two columns. The stems are listed down in the left column (from -0 to 2).

- Separate each rounded data value into a stem (of thousands) and a leaf (of hundreds).

The first data value, -400, the stem is -0 as it has no number in the thousands place and 4 is the leaf as it has 4 in the hundreds place. Write 4 in the row of -0 stem.

The next data value, -200, the stem is -0 as it has no number in the thousands place and 2 is the leaf as it has 2 in the hundreds place. Write 2 in the row of -0 stem.

The next data value, -0, the stem is -0 and 0 is the leaf. Write 0 in the row of -0 stem.

The next data value, -0, the stem is -0 and 0 is the leaf. Write 0 in the row of -0 stem.

Key: -0|4 = -400

- The -0 stem includes the (rounded) values from -900 to -0.
- The 0 stem includes the values from 0 to 900.
- The 1 stem includes the values from 1000 to 1900.
- The 2 stem includes the values from 2000 to 2900.

## Types of stem and leaf plots

**Simple stem plots**

All the above examples are simple stem and leaf plots. In these plots, the stem values are repeated once, no matter how many leaves it contains.

The following is a stem and leaf plot of the heights in cm of 30 participants in a certain survey.

Here is the raw data

147 150 153 155 155 155 156 156 156 157

158 159 160 160 160 160 161 162 163 163

163 164 167 167 169 170 172 174 180 180

Here is the stem and leaf plot

Key: 14|7 means 147 cm.

When the leaves are too crowded, it may be desired to use split stem and leaf plots.

**Split stem and leaf plots**

Where each stem is split into two equal parts. This may show additional patterns in our data distribution.

For the above example of heights, the following is the split stem and leaf plot for the same data.

Key: 14|7 means 147 cm.

- The first 14 stem includes the values from 140 to 144.
- The second 14 stem includes the values from 145 to 149.
- The first 15 stem includes the values from 150 to 154.
- The second 15 stem includes the values from 155 to 159.
- The first 16 stem includes the values from 160 to 164.
- The second 16 stem includes the values from 165 to 169, and so on.
- In the first simple stem and leaf plot, we can conclude that the main cluster of data is between 150 to 169 cm.
- But in the split stem and leaf plot, we can conclude that the main cluster of data is between 155 to 164 cm which is a more accurate conclusion.

**Back-to-back stem and leaf plots**

These are used to compare the distribution of numerical values across two groups.

The following is the heights in cm of 20 male participants in a survey

155 156 156 160 162 162 163 164 165 167

167 167 169 169 170 170 172 174 174 178

The following is the heights in cm of 20 female participants in a survey

147 150 153 155 155 156 157 158 158 158

159 159 160 160 160 160 161 163 163 165

Here is a back-to-back stem and leaf plot comparing males to females

Key: 14|7 = 147 cm, 8|17 = 178 cm.

- The stem represents tens and leaves represent ones.
- The right-most column is for the female leaves and the left-most column is for the male leaves.
- The leaves in the right column are arranged in ascending order, while the leaves in the left column are arranged in descending order.

We can also split the stems to improve the visualization

We can conclude that:

- The minimum height for males is 155 cm and the maximum height is 178 cm.
- The minimum height for females is 147 cm and the maximum height is 165 cm.
- The heights of females are clustered at 155-164 cm, while the heights of males are clustered at 160-174 cm.

**Preguntas practicas**

- The following is a stem and leaf plot of the weights of 20 persons

Key: 8|0 = 80 kg.

How many persons have a weight = 69 kg?

- The following is the stem and leaf plot of the systolic blood pressure of 15 persons

Key: 16|8 = 168.

How many persons have a blood pressure = 140?

What is the maximum and minimum of this data?

- The following is the data and stem and leaf plot for 15 persons’ balance account.

Here is the raw data

2143, 29, 2, 1506, 1, 231, 447, 2, 121, 593, 270, 390, 6, 71, 162

Here is the stem and leaf plot

Key: 21|4 = 2140

Why 2140 is present although it is not in the raw data?

Why several zeros appear in the first row, although none of the persons has zero balance?

- The following is the stem and leaf plot of 14 ozone measurements

Key: 46|0 = 46.0

How you can improve this plot?

- Here is a back-to-back stem and leaf plot comparing scores for two classes. Each class has 20 students.

Key: 4|7 = 47.

Which class has the maximum score, which class has the minimum score?

**respuestas**

- The stems represent tens and the leaves ones. We look at stem 6 and count the number of 9 leaves. There are three 9 leaves in the 6 stem row so 3 persons are weighing 69 kg.
- The stems represent tens and the leaves ones. We look at stem 14 and count the number of 0 leaves. There are no 0 leaves in the 14 stem row so no persons are having a systolic blood pressure = 140 in this data.

We look at the first stem row to detect the minimum. This is a split stem and leaf plot. The first 9 stem row is empty meaning that there are no values in the range 90-94.

The second row contains the 5 leaf in the 9 stem so the minimum = 95.

We look at the last row to get the maximum. The last row contains 8 leaf in the 16 stem so the maximum = 168.

- By looking at the key, 21|4 = 2140, we see that stems represent hundreds and the leaves tens so the raw data are rounded to the nearest tens.

The value 2143 is rounded to 2140 so it is shown in the stem plot although it is not present in the raw data.

The 3 zeros in the first row represent the data values that are less than 5 and it is rounded to 0. These values are 1,2,2.

- The provided stem and leaf plot show the decimal places as leaves and stems as ones and tens. It is running from a minimum of 6 to a maximum of 46 or 41 rows and it is difficult to read.

We can improve this plot by setting the stems as tens and leaves as ones. So the stem plot will run from 0 to 4 or 5 rows only.

Key: 4|6 = 46.

- Look at the first row to see the minimum for every class.

The minimum of class 1 is 47 and the minimum of class 2 is 55.

Class 1 has the minimum score.

Look at the last row to see the maximum for every class.

The maximum of class 1 is 89 and the maximum of class 2 is 87.

Class 1 has the maximum score.