How To Calculate Iqr In R
What is the Interquartile Range (IQR)?
The interquartile range (IQR) measures the spread of the centre half of your data. Information technology is the range for the eye 50% of your sample. Use the IQR to assess the variability where nigh of your values lie. Larger values point that the central portion of your data spread out farther. Conversely, smaller values show that the middle values cluster more tightly.
In this post, learn what the interquartile range ways and the many ways to utilise it! I'll show you how to find the interquartile range, apply it to measure variability, graph it in boxplots to assess distribution properties, use it to identify outliers, and exam whether your data are normally distributed.
The interquartile range is one of several measures of variability. To learn near the others and how the IQR compares, read my post, Measures of Variability.
Interquartile Range Definition
To visualize the interquartile range, imagine dividing your data into quarters. Statisticians refer to these quarters every bit quartiles and characterization them from low to high as Q1, Q2, Q3, and Q4. The lowest quartile (Q1) covers the smallest quarter of values in your dataset. The upper quartile (Q4) comprises the highest quarter of values. The interquartile range is the middle half of the data that lies between the upper and lower quartiles. In other words, the interquartile range includes the 50% of data points that are above Q1 and below Q4. The IQR is the red area in the graph below, containing Q2 and Q3 (not labeled).
When measuring variability, statisticians adopt using the interquartile range instead of the full data range because extreme values and outliers affect information technology less. Typically, use the IQR with a measure of primal trend, such as the median, to understand your data's center and spread. This combination creates a fuller picture of your information'south distribution.
Dissimilar the more familiar hateful and standard difference, the interquartile range and the median are robust measures. Outliers do not strongly influence either statistic because they don't depend on every value. Additionally, like the median, the interquartile range is superb for skewed distributions. For normal distributions, you can use the standard divergence to determine the percent of observations that fall specific distances from the mean. However, that doesn't work for skewed distributions, and the IQR is an first-class alternative.
Related posts: Quartiles: Definition, Finding, and Using, Median: Definition and Uses, and What are Robust Statistics?
How to Observe the Interquartile Range (IQR) past Hand
The formula for finding the interquartile range takes the third quartile value and subtracts the first quartile value.
IQR = Q3 – Q1
Equivalently, the interquartile range is the region between the 75th and 25th percentile (75 – 25 = l% of the data).
Using the IQR formula, we need to find the values for Q3 and Q1. To exercise that, but society your data from low to high and split the value into four equal portions.
I've divided the dataset beneath into quartiles. The interquartile range extends from the Q1 value to the Q3 value. For this dataset, the interquartile range is 39 – 20 = 19.
Note that different methods and statistical software programs will find slightly different Q1 and Q3 values, which affects the interquartile range. These variations stem from alternate ways of finding percentiles. For details about that, read my post about Percentiles: Interpretations and Calculations.
How to Find the Interquartile Range using Excel
All statistical software packages volition identify the interquartile range as role of their descriptive statistics. Hither, I'll show you how to find information technology using Excel because most readers can access this application.
To follow along, download the Excel file: IQR. This dataset is the same as the 1 I use in the illustration in a higher place. This file also includes the interquartile range calculations for finding outliers and the IQR normality examination described later in this post.
In Excel, you'll need to employ the QUARTILE.EXC part, which has the post-obit arguments: QUARTILE.EXC(array, quart)
- Array: Cell range of numeric values.
- Quart: Quartile you want to find.
In my spreadsheet, the data are in cells A2:A20. Consequently, I'll use the following syntax to find Q1 and Q3, respectively:
- =QUARTILE.EXC(A2:A20,one)
- =QUARTILE.EXC(A2:A20,3)
As with my example of finding the interquartile range past mitt, Excel indicates that Q3 is 39 and Q1 is 20. IQR = 39 – xx = 19
Related post: Descriptive Statistics in Excel
Using Boxplots to Graph the Interquartile Range
Boxplots are a keen way to visualize interquartile ranges and their relation to the median and the overall distribution. These graphs display ranges of values based on quartiles and bear witness asterisks for outliers that fall outside the whiskers. Boxplots piece of work by splitting your data into quarters.
Let's look at the boxplot beefcake before getting to the example. Observe how it divides your data into quartiles.
The box in the boxplot is your interquartile range! It contains 50% of your data. By comparison the size of these boxes, y'all tin can understand your data'south variability. More dispersed distributions have wider boxes.
Additionally, notice where the median line falls within each interquartile box. If the median is closer to one side or the other of the box, information technology's a skewed distribution. When the median is about the middle of the interquartile range, your distribution is symmetric.
For instance, in the boxplot below, method iii has the highest variability in scores and is left-skewed. Conversely, method 2 has a tighter distribution that is symmetrical, although it also has an outlier—read the next section for more about that!
Related postal service: Boxplots versus Individual Value Plots
Using the IQR to Find Outliers
The interquartile range tin assistance yous place outliers. For other methods of finding outliers, the outliers themselves influence the calculations, potentially causing you to miss them. Fortunately, interquartile ranges are relatively robust against outlier influence and tin can avoid this problem. This method also does not presume the data follow the normal distribution or any other distribution. That's why using the IQR to find outliers is 1 of my favorite methods!
To find outliers, you'll need to know your data'southward IQR, Q1, and Q3 values. Take these values and input them into the equations below. Statisticians call the effect for each equation an outlier gate. I've included these calculations in the IQR example Excel file.
Q1 − one.5 * IQR: Lower outlier gate.
Q3 + 1.5 * IQR: Upper outlier gate.
Using the aforementioned example dataset, I'll calculate the two outlier gates. For that dataset, the interquartile range is nineteen, Q1 = 20, and Q3 = 39.
Lower outlier gate: 20 – one.v * xix = -8.five
Upper outlier gate: 39 + one.5 * 19 = 67.v
Then wait for values in the dataset that are below the lower gate or above the upper gate. For the example dataset, there are no outliers. All values fall betwixt these two gates.
Boxplots typically use this method to identify outliers and display asterisks when they exist. In the teaching method boxplot above, observe that the Method two group has an outlier. The researchers should investigate that value.
Related mail: Five Ways to Find Outliers
Using the Interquartile Range to Test Normality
You can even employ the interquartile range every bit a simple test to determine whether your data are normally distributed. When data follow a normal distribution, the interquartile range will have specific backdrop. The prototype beneath highlights these properties. Specifically, in our calculations below, we'll employ the standard deviations (σ) that represent to the interquartile range, -0.67 and 0.67.
You can assess whether your IQR is consequent with a normal distribution. However, this test should not replace a formal normality hypothesis test.
To perform this test, y'all'll demand to know the sample standard divergence (s) and sample hateful (x̅). Input these values into the formulas for Q1 and Q3 below.
- Q1 = x̅ − (s * 0.67)
- Q3 = x̅ + (south * 0.67)
Compare these calculated values to your data'due south bodily Q1 and Q3 values. If they are notably dissimilar, your data might non follow the normal distribution.
Nosotros'll render to our example dataset from before. Our actual Q1 and Q3 are twenty and 39, respectively.
The sample boilerplate is 31.3, and its standard deviation is xiv.i. I'll input those values into the equations.
Q1 = 31.3 – (xiv.ane * 0.67) = 21.nine
Q3 = 31.3 + (14.1 * 0.67) = 40.7
The calculated values are pretty close to the actual information values, suggesting that our data follow the normal distribution. I've included these calculations in the IQR example spreadsheet.
Related posts: Agreement the Normal Distribution and How to Identify the Distribution of Your Data
Source: https://statisticsbyjim.com/basics/interquartile-range/
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