Five Number Summary on Calculator

A five-number summary is a simple and effective way to describe the distribution of a dataset using five key values:

1. Minimum
2. First Quartile (Q1)
3. Median (Q2)
4. Third Quartile (Q3)
5. Maximum

Together, these five values provide a quick overview of the center, spread, and overall distribution of a dataset. The five-number summary is also the foundation of a box-and-whisker plot.

What Is a Five-Number Summary?

The five-number summary consists of:

• Minimum: The smallest observation in the dataset.
• Q1: The 25th percentile; approximately 25% of observations are at or below this value.
• Median: The 50th percentile; it divides the ordered dataset into two halves.
• Q3: The 75th percentile; approximately 75% of observations are at or below this value.
• Maximum: The largest observation in the dataset.

It is particularly useful when you want to summarize a dataset without relying only on the mean and standard deviation.

Five Number Summary on a TI-84 Calculator

Let’s use the following dataset:

4, 6, 6, 7, 8, 12, 15, 17, 20, 21, 21, 23, 24, 27, 28

We will use a TI-84 calculator to obtain the five-number summary.

Step 1: Enter the Data

First, enter the observations into the calculator.

1. Press STAT.
2. Select 1:Edit.
3. Enter the values into L1.

Your list should contain:

4667812151720212123242728

Make sure all observations have been entered correctly before continuing.

Step 2: Open 1-Var Stats

Next, calculate the descriptive statistics.

1. Press STAT.
2. Move to the CALC menu.
3. Select 1-Var Stats.
4. Press ENTER.
5. If necessary, enter L1 as the list.
6. Press ENTER.

The TI-84 will display a statistical summary of the dataset.

Step 3: Find the Five-Number Summary

For this dataset, the five-number summary is:

Therefore, the five-number summary can be written as:

(4, 7, 17, 23, 28)

These five values provide a compact description of the distribution.

Understanding Each Value

1. Minimum = 4

The minimum is the smallest value in the dataset.

In this example:

Minimum = 4

2. First Quartile (Q1) = 7

Q1 represents the lower quartile. It indicates the point below which approximately 25% of the observations fall.

Here:

Q1 = 7

3. Median = 17

The median is the middle value of the ordered dataset.

Because there are 15 observations, the eighth observation is the median:

Median = 17

4. Third Quartile (Q3) = 23

Q3 represents the upper quartile. Approximately 75% of observations fall at or below Q3.

Here:

Q3 = 23

5. Maximum = 28

The maximum is the largest observation:

Maximum = 28

Range From the Five-Number Summary

The range measures the total spread between the smallest and largest observations.

The formula is:

Range = Maximum − Minimum

For our dataset:

Range = 28 − 4

Range = 24

Therefore, the total range of the dataset is 24.

Interquartile Range (IQR)

The interquartile range (IQR) measures the spread of the middle 50% of observations.

The formula is:

IQR = Q3 − Q1

For our example:

IQR = 23 − 7

IQR = 16

Therefore, the middle 50% of the observations span 16 units.

Five-Number Summary and Box Plot

The five-number summary is directly related to a box-and-whisker plot.

The structure is:

Minimum → Q1 → Median → Q3 → Maximum

For our example:

4 → 7 → 17 → 23 → 28

The box extends from Q1 = 7 to Q3 = 23, with the median at 17. The whiskers extend toward the minimum and maximum, subject to the outlier convention used when constructing the box plot.

This makes the five-number summary particularly useful for visually comparing multiple datasets.

Using the Five-Number Summary to Understand Distribution

The five-number summary can provide useful information about the shape and spread of a dataset.

For example, compare the distances:

• Minimum to Q1: 7 − 4 = 3
• Q1 to Median: 17 − 7 = 10
• Median to Q3: 23 − 17 = 6
• Q3 to Maximum: 28 − 23 = 5

These intervals can provide clues about whether the distribution is approximately symmetric or potentially skewed.

However, the five-number summary alone does not provide the complete shape of the distribution. A histogram or box plot can provide additional information.

Identifying Potential Outliers

The five-number summary can also be used to identify potential outliers using the 1.5 × IQR rule.

First calculate the IQR:

IQR = 16

Then calculate the lower fence:

Lower Fence = Q1 − 1.5 × IQR

= 7 − 1.5(16)

= −17

The upper fence is:

Upper Fence = Q3 + 1.5 × IQR

= 23 + 1.5(16)

= 47

Therefore, values below −17 or above 47 would be considered potential outliers under this rule.

The minimum and maximum values in this example, 4 and 28, fall within these boundaries.

Why Use a Five-Number Summary?

A five-number summary is useful because it provides several important pieces of information using only five values.

It can help you:

• Understand the center of a dataset
• Measure overall spread
• Examine the middle 50% of observations
• Compare multiple datasets
• Identify potential outliers
• Construct box-and-whisker plots
• Understand possible skewness

It is especially useful when the data are not normally distributed or when extreme values may make the mean and standard deviation less representative.

Five-Number Summary vs. Mean and Standard Deviation

The five-number summary and the mean/standard deviation provide different types of information.

For skewed data or data containing outliers, the median and IQR can often provide a more robust description of the distribution than the mean and standard deviation.

Example: Interpreting a Five-Number Summary

Suppose a dataset has the following five-number summary:

(10, 18, 25, 32, 60)

We can immediately determine that:

• The minimum is 10.
• Q1 is 18.
• The median is 25.
• Q3 is 32.
• The maximum is 60.
• The range is 50.
• The IQR is 14.

The relatively large distance between Q3 and the maximum may indicate that the upper end of the distribution deserves further investigation.

Frequently Asked Questions

What are the five numbers in a five-number summary?

The five numbers are:

Minimum, Q1, Median, Q3, and Maximum.

What is the formula for the five-number summary?

There is no single formula. The five-number summary is a collection of five descriptive statistics:

Minimum, Q1, Median, Q3, Maximum

What is the difference between Q1 and Q3?

Q1 is the 25th percentile, while Q3 is the 75th percentile.

The difference between them is the interquartile range:

IQR = Q3 − Q1

What does the five-number summary tell you?

It provides a compact description of the center, spread, and distribution of a dataset.

Can a five-number summary identify outliers?

It can be used to identify potential outliers when combined with the 1.5 × IQR rule.

Conclusion

The five-number summary is one of the most useful ways to quickly describe a dataset. It consists of the minimum, first quartile (Q1), median, third quartile (Q3), and maximum.

For the dataset:

4, 6, 6, 7, 8, 12, 15, 17, 20, 21, 21, 23, 24, 27, 28

the five-number summary is:

(4, 7, 17, 23, 28)

From these five values, we can also calculate:

Range = 28 − 4 = 24

IQR = 23 − 7 = 16

The TI-84 calculator makes it convenient to calculate descriptive statistics and analyze datasets. Once you understand how to interpret the five-number summary, you can use it to compare distributions, construct box plots, investigate variability, and identify potential outliers.