Frequency is the number of times a specific data value occurs in your dataset. A frequency table lists a set of values and how often each one appears. They help you understand which data values are common and which are rare. These tables organize your data and are an effective way to present the results to others. Frequency tables are also known as frequency distributions because they allow you to understand the distribution of values in your dataset. [Read more…] about Frequency Table
The mean absolute deviation (MAD) is a measure of variability that indicates the average distance between observations and their mean. MAD uses the original units of the data, which simplifies interpretation. Larger values signify that the data points spread out further from the average. Conversely, lower values correspond to data points bunching closer to it. The mean absolute deviation is also known as the mean deviation and average absolute deviation. [Read more…] about Mean Absolute Deviation
A conditional probability is the likelihood of an event occurring given that another event has already happened. Conditional probabilities allow you to evaluate how prior information affects probabilities. When you incorporate existing facts into the calculations, it can change the probability of an outcome. [Read more…] about Conditional Probability
Cluster sampling is a method of obtaining a representative sample from a population that researchers have divided into groups. An individual cluster is a subgroup that mirrors the diversity of the whole population while the set of clusters are similar to each other. Typically, researchers use this approach when studying large, geographically dispersed populations because it is a cost-controlling measure. [Read more…] about Cluster Sampling
Stratified sampling is a method of obtaining a representative sample from a population that researchers have divided into relatively similar subpopulations (strata). Researchers use stratified sampling to ensure specific subgroups are present in their sample. It also helps them obtain precise estimates of each group’s characteristics. Many surveys use this method to understand differences between subpopulations better. Stratified sampling is also known as stratified random sampling. [Read more…] about Stratified Sampling
A skewed distribution occurs when one tail is longer than the other. Skewness defines the asymmetry of a distribution. Unlike the familiar normal distribution with its bell-shaped curve, these distributions are asymmetric. The two halves of the distribution are not mirror images because the data are not distributed equally on both sides of the distribution’s peak. [Read more…] about Skewed Distribution
Heterogeneity is defined as a dissimilarity between elements that comprise a whole. When heterogeneity is present, there is diversity in the characteristic under study. The parts of the whole are different, not the same. It is an essential concept in science and statistics. Heterogeneous is the opposite of homogeneous. [Read more…] about Heterogeneity
Control variables are properties that researchers hold constant for all observations in an experiment. While these variables are not the primary focus of the research, keeping their values consistent helps the study establish the true relationships between the independent and dependent variables. Control variables are different from control groups. [Read more…] about Control Variables
Orthogonality is a mathematical property that is beneficial for statistical models. It’s particularly helpful when performing factorial analysis of designed experiments. [Read more…] about Orthogonality
Accuracy and precision are crucial properties of your measurements when you’re relying on data to draw conclusions. Both concepts apply to a series of measurements from a measurement system.
Measurement systems facilitate the quantification of characteristics for data collection. They include a collection of instruments, software, and personnel necessary to assess the property of interest. For example, a research project studying bone density will devise a measurement system to produce accurate and precise measurements of bone density. [Read more…] about Accuracy vs Precision
A control group in an experiment does not receive the treatment. Instead, it serves as a comparison group for the treatments. Researchers compare the results of a treatment group to the control group to determine the effect size, also known as the treatment effect. [Read more…] about Control Group in an Experiment
The range of a data set is the difference between the maximum and the minimum values. It measures variability using the same units as the data. Larger values represent greater variability.
The range is the easiest measure of dispersion to calculate and interpret in statistics, but it has some limitations. In this post, I’ll show you how to find the range mathematically and graphically, interpret it, explain its limitations, and clarify when to use it. [Read more…] about Range of a Data Set
A z-score measures the distance between a data point and the mean using standard deviations. Z-scores can be positive or negative. The sign tells you whether the observation is above or below the mean. For example, a z-score of +2 indicates that the data point falls two standard deviations above the mean, while a -2 signifies it is two standard deviations below the mean. A z-score of zero equals the mean. Statisticians also refer to z-scores as standard scores, and I’ll use those terms interchangeably. [Read more…] about Z-score: Definition, Formula, and Uses
Pascal’s triangle is a number pattern that fits in a triangle. It is named after Blaise Pascal, a French mathematician, and it has many beneficial mathematic and statistical properties, including finding the number of combinations and expanding binomials. [Read more…] about Pascal’s Triangle
Robust statistics provide valid results across a broad variety of conditions, including assumption violations, the presence of outliers, and various other problems. The term “robust statistic” applies both to a statistic (i.e., median) and statistical analyses (i.e., hypothesis tests and regression). [Read more…] about What are Robust Statistics?
A relative frequency indicates how often a specific kind of event occurs within the total number of observations. It is a type of frequency that uses percentages, proportions, and fractions.
In this post, learn about relative frequencies, the relative frequency distribution, and its cumulative counterpart. [Read more…] about Relative Frequencies and Their Distributions
Venn diagrams visually represent relationships between concepts. They use circles to display similarities and differences between sets of ideas, traits, or items. Intersections indicate that the groups have common elements. Non-overlapping areas represent traits that are unique to one set. Venn diagrams are also known as logic diagrams and set diagrams. [Read more…] about Venn Diagrams: Uses, Examples, and Making
The empirical rule in statistics, also known as the 68-95-99.7 rule, states that for normal distributions, 68% of observed data points will lie inside one standard deviation of the mean, 95% will fall within two standard deviations, and 99.7% will occur within three standard deviations. [Read more…] about Empirical Rule: Definition, Formula, and Uses
The interquartile range (IQR) measures the spread of the middle half of your data. It is the range for the middle 50% of your sample. Use the IQR to assess the variability where most of your values lie. Larger values indicate that the central portion of your data spread out further. Conversely, smaller values show that the middle values cluster more tightly.
In this post, learn what the interquartile range means and the many ways to use it! I’ll show you how to find the interquartile range, use it to measure variability, graph it in boxplots to assess distribution properties, use it to identify outliers, and test whether your data are normally distributed.
The interquartile range is one of several measures of variability. To learn about the others and how the IQR compares, read my post, Measures of Variability.
Interquartile Range Overview
To visualize the interquartile range, imagine dividing your data into quarters. Statisticians refer to these quarters as quartiles and label 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 prefer using the interquartile range instead of the full data range because extreme values and outliers affect it less. Typically, use the IQR with a measure of central tendency, such as the median, to understand your data’s center and spread. This combination creates a fuller picture of your data’s distribution.
Unlike the more familiar mean and standard deviation, 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 deviation to determine the percentage of observations that fall specific distances from the mean. However, that doesn’t work for skewed distributions, and the IQR is an excellent alternative.
How to Find the IQR by Hand
The formula for calculating 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 = 50% of the data).
Using the IQR formula, we need to find the values for Q3 and Q1. To do that, simply order your data from low to high and split the value into four equal portions.
I’ve divided the dataset below 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.
Finding the Interquartile Range using Excel
All statistical software packages will identify the interquartile range as part of their descriptive statistics. Here, I’ll show you how to find it using Excel because most readers can access this application.
To follow along, download the Excel file: IQR. This dataset is the same as the one I use in the illustration above. This file also includes the interquartile range calculations for finding outliers and the IQR normality test described later in this post.
In Excel, you’ll need to use the QUARTILE.EXC function, which has the following 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:
As with my example of finding the interquartile range by hand, Excel indicates that Q3 is 39 and Q1 is 20. IQR = 39 – 20 = 19
Related post: Descriptive Statistics in Excel
Using Boxplots to Graph the Interquartile Range
Boxplots are a great 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 show asterisks for outliers that fall outside the whiskers. Boxplots work by splitting your data into quarters.
Let’s look at the boxplot anatomy before getting to the example. Notice how it divides your data into quartiles.
The box in the boxplot is your interquartile range! It contains 50% of your data. By comparing the size of these boxes, you can understand your data’s variability. More dispersed distributions have wider boxes.
Additionally, find where the median line falls within each interquartile box. If the median is closer to one side or the other of the box, it’s a skewed distribution. When the median is near the center of the interquartile range, your distribution is symmetric.
For example, in the boxplot below, method 3 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 post: Boxplots versus Individual Value Plots
Using the IQR to Find Outliers
The interquartile range can help you identify 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 can avoid this problem. This method also does not assume the data follow the normal distribution or any other distribution. That’s why using the IQR to find outliers is one of my favorite methods!
To find outliers, you’ll need to know your data’s IQR, Q1, and Q3 values. Take these values and input them into the equations below. Statisticians call the result for each equation an outlier gate. I’ve included these calculations in the IQR example Excel file.
Q1 − 1.5 * IQR: Lower outlier gate.
Q3 + 1.5 * IQR: Upper outlier gate.
Using the same example dataset, I’ll calculate the two outlier gates. For that dataset, the interquartile range is 19, Q1 = 20, and Q3 = 39.
Lower outlier gate: 20 – 1.5 * 19 = -8.5
Upper outlier gate: 39 + 1.5 * 19 = 67.5
Then look 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 between these two gates.
Boxplots typically use this method to identify outliers and display asterisks when they exist. In the teaching method boxplot above, notice that the Method 2 group has an outlier. The researchers should investigate that value.
Related post: Five Ways to Find Outliers
Using the Interquartile Range to Test Normality
You can even use the interquartile range as a simple test to determine whether your data are normally distributed. When data follow a normal distribution, the interquartile range will have specific properties. The image below highlights these properties. Specifically, in our calculations below, we’ll use the standard deviations (σ) that correspond to the interquartile range, -0.67 and 0.67.
You can assess whether your IQR is consistent with a normal distribution. However, this test should not replace a formal normality hypothesis test.
To perform this test, you’ll need to know the sample standard deviation (s) and sample mean (x̅). Input these values into the formulas for Q1 and Q3 below.
- Q1 = x̅ − (s * 0.67)
- Q3 = x̅ + (s * 0.67)
Compare these calculated values to your data’s actual Q1 and Q3 values. If they are notably different, your data might not follow the normal distribution.
We’ll return to our example dataset from before. Our actual Q1 and Q3 are 20 and 39, respectively.
The sample average is 31.3, and its standard deviation is 14.1. I’ll input those values into the equations.
Q1 = 31.3 – (14.1 * 0.67) = 21.9
Q3 = 31.3 + (14.1 * 0.67) = 40.7
The calculated values are pretty close to the actual data values, suggesting that our data follow the normal distribution. I’ve included these calculations in the IQR example spreadsheet.