Quartiles and quantiles are statistical units used in quantitative research. Quartiles and quantiles are applied in data science and analysis to learn about spending, average scores, and other real-life deductions from data.^{1}

## Definition: Quartiles and quantiles

**Quartiles** are a set of three values that divide ordered data sets into four equal parts. **Quantiles** are units of equal, adjacent quantities in a distribution. Quartiles, percentiles, and deciles are examples of quantiles used in research as statistical quantities.^{2}

## What are quartiles?

Quartiles are mathematical values that split a set of observations into four intervals, i.e., quarters.

The **first quartile**, denoted as **Q1**, is also known as the lower quartile. It is the 25th percentile in a data distribution, implying that 25% of the units in the data set are located below the first quartile.

The **second quartile** is the median of the distribution denoted as **Q2**. It is the 50th percentile in the distribution, i.e., 50% of the quantities are located below the median.

The **third quartile**, **Q3**, is the upper quartile. It is the distribution’s 75th percentile, so 75% of the quantities are located below the upper quartile.

Dividing an ordered distribution into **Q1, Q2,** and **Q3**, the data is split into four equal parts. Quartiles also split the range of a probability distribution into four equal probability intervals.^{3}

## How to find quartiles and quantiles

Quartiles and quantiles are calculated using mathematical formulas to derive distribution characteristics. Quartiles can be calculated as follows:

- Find the number of units in the distribution (n)

- Arrange the observations from the smallest to the largest

- Calculate the first quartile (Q1)

- Find .
- Where is an integer (whole number),
**Q1**is the average of the numbers that fall between and . - Where is not a whole number, you can round it up to the nearest whole number. The number that falls in this position is
**Q1**or the first quartile.

- Calculate the second quartile (Q2)

- Find .
- If the result is an integer,
**Q2**(the second quartile) is the mean of the numbers that fall between and . - Where is not a whole number, you should round it up; the number in this position is
**Q2**, i.e., the second quartile.

- Calculate the third quartile (Q3)

- Find .
- Where the result is an integer,
**Q3**is the mean of the numbers that fall between and . - If is not a whole number, round up your result. The third quartile is the number that falls in this position.

**Step 1:** Find the total number of observations (n)

n = 2+2+2+2 = 8

**Step 2:** Arrange the observations from the smallest to the largest

1,1,2,2,3,3,5,5

**Step 3:** Determine the first quartile

= = 2

Q1 is the number at the 2nd position = **1**

**Step 4:** Determine the second quartile (Q2)

= = 4

Q2 is the number at the 4th position is = **2**

**Step 5:** Calculate the third quartile

= = 6

Q3 is the number at the 6th position = **3**

## Visualizing quartiles with boxplots

Quartiles and quantiles can be visualized using boxplots. To understand quartiles and quantiles, statisticians use visual representations to draw conclusions more accurately.

Use a five-number summary to make a boxplot as follows:

- Find the minimum – This is the lowest number in an ordered distribution. It is also known as the zeroth quantile.
- Find the first quartile.
- Find the second quartile
- Find the third quartile.
- Find the maximum- The highest number in a set of observations.
^{4}

## Interpreting quartiles and quantiles

Quartiles and quantiles can provide useful insights from a data set. They can be interpreted using the following methods:

### Comparing observations

Quartiles and quantiles are used in statistics to draw comparisons. Quartiles show the characteristics of a sample from a statistical standpoint.

### The median

Quartiles and quantiles are also used to find the median. The median is the midpoint of an ordered distribution and is the second quartile in the distribution.^{5}

### Interquartile range

In quartile and quantile analysis, the interquartile range is the difference between the first and third quartile, i.e., **IQR = Q3 – Q1**. It is a statistical measure of variability used to determine the spread of the mid-50% of the distribution.

### Skewness

We use skewness in quartile and quantile calculations to determine symmetry or skewness in a given sample. Skewness refers to the deviation of data from a normal distribution, i.e., symmetry. Boxplots are mainly used to visualize skewness and symmetry.^{6}

### Highlighting outliers

Outliers in quartile and quantile exploration are numbers that are too large or too small relative to the rest of the observations. Outliers in statistics often distort the interquartile range.

## What are quantiles?

**Quantiles** is an umbrella term for quartiles and other measures of distribution. Quartiles and quantiles may be used interchangeably when a researcher calculates quartiles and the interquartile range. Quantiles divide ordered data samples into equal units. Examples of quantiles include:

**Quartiles –** Four equal parts divided by three quartiles.

**Deciles –** Ten parts divided by nine percentiles.

**Percentiles –** A hundred parts divided by 99 percentiles.

**How to Find Quantiles**

Quartiles and quantiles are calculated using similar methods, but quantiles are multiplied by instead of in the last three steps as follows:

**Example**:

If you’re calculating the second 5-quartile in a distribution:

- Find .
- Where the result is an integer, the second 5-quartile is the average of the numbers between and .
- Where is a non-integer, round up the result to get the second 5-quartile in that position.

## FAQs

Quartiles and quantiles are statistical metrics calculated to determine the characteristics of a distribution. Quartiles are units used to divide data into four equal segments, while quantiles include quartiles and other measures of data division.

Quartiles and quantiles are used in quantitative research. They show the median, skewness, and symmetrical characteristics of data samples.

Quartiles and quantiles can be calculated manually for small samples using n multiplied by the **Q1, Q2,** and **Q3** multiples. Statistical calculators also have n-built functions to determine quartiles and quantiles.

Quartiles and quantiles can be calculated using python, R, and other powerful programs. These tools can quickly calculate quartiles and quantiles for large data groups such as health records and disease progression.

## Sources

^{1} MathIsFun. “Quartiles.” Accessed August 18, 2022. https://www.mathsisfun.com/data/quartiles.html.

^{2} Liberto, Daniel. “Quartile.” Investopedia. October 07, 2021. https://www.investopedia.com/terms/q/quartile.asp.

^{3 }Glen, Stephanie. “What are Quartiles?.” StatisticsHowTo. Accessed August 18, 2022. https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/what-are-quartiles/.

^{4 }Khan Academy. “What is a box and whisker plot?.” Accessed August 18, 2022. https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/a/box-plot-review.

^{5 } MathisFun. “How to Find the Median Value.” Accessed August 19, 2022. https://www.mathsisfun.com/median.html.

^{6 } CFI Team. “Skewness.” May 06, 2022. https://corporatefinanceinstitute.com/resources/knowledge/other/skewness/.