Making statistics intuitive
The lognormal distribution is a continuous probability distribution that models right-skewed data. The unimodal shape of the lognormal distribution is comparable to the Weibull and loglogistic distributions. [Read more…] about Lognormal Distribution: Uses, Parameters & Examples
In the United States, our Thanksgiving holiday is fast approaching. On this day, we give thanks for the good things in our lives.
For this post, I wanted to quantify how thankful we should be. Ideally, I’d quantify something truly meaningful, like happiness. Unfortunately, most countries are not like Bhutan, which measures the gross national happiness and incorporates it into their five-year development plans.
Instead, I’ll focus on something that is more concrete and regularly measured around the world—income. By examining income distributions, I’ll show that you have much to be thankful for, and so does most of the world! [Read more…] about A Statistical Thanksgiving: Global Income Distributions
Variance is a measure of variability in statistics. It assesses the average squared difference between data values and the mean. Unlike some other statistical measures of variability, it incorporates all data points in its calculations by contrasting each value to the mean. [Read more…] about Variance: Definition, Formulas & Calculations
The uniform distribution is a symmetric probability distribution where all outcomes have an equal likelihood of occurring. All values in the distribution have a constant probability, making them uniformly distributed. This distribution is also known as the rectangular distribution because of its shape in probability distribution plots, as I’ll show you below. [Read more…] about Uniform Distribution: Definition & Examples
A frequency table lists a set of values and how often each one appears. Frequency is the number of times a specific data value occurs in your dataset. These tables 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: How to Make & Examples
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: Definition, Finding & Formula
Stem and leaf plots display the shape and spread of a continuous data distribution. These graphs are similar to histograms, but instead of using bars, they show digits. It’s a particularly valuable tool during exploratory data analysis. They can help you identify the central tendency, variability, skewness of your distribution, and outliers. Stem and leaf plots are also known as stemplots. [Read more…] about Stem and Leaf Plot: Making, Reading & Examples
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: Definition & Examples
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
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
The empirical rule in statistics, also known as the 68 95 99 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
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.
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 lower quartile (Q1) divides the smallest quarter of values in your dataset from the higher values. The upper quartile (Q3) is the value that separates the highest quarter of values from the rest of the dataset. 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 Q3. The IQR is the red area in the graph below.
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.
Related posts: Quartiles: Definition, Finding, and Using, Median: Definition and Uses, and What are Robust Statistics?
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 = 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.
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)
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
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: Box Plots Explained with Examples
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
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.
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.
Related posts: Understanding the Normal Distribution and How to Identify the Distribution of Your Data
The standard deviation (SD) is a single number that summarizes the variability in a dataset. It represents the typical distance between each data point and the mean. Smaller values indicate that the data points cluster closer to the mean—the values in the dataset are relatively consistent. Conversely, higher values signify that the values spread out further from the mean. Data values become more dissimilar, and extreme values become more likely. [Read more…] about Standard Deviation: Interpretations and Calculations
The mean in math and statistics summarizes an entire dataset with a single number representing the data’s center point or typical value. It is also known as the arithmetic mean, and it is the most common measure of central tendency. It is frequently called the “average.” [Read more…] about What is the Mean and How to Find It: Definition & Formula
The gamma distribution is a continuous probability distribution that models right-skewed data. Statisticians have used this distribution to model cancer rates, insurance claims, and rainfall. Additionally, the gamma distribution is similar to the exponential distribution, and you can use it to model the same types of phenomena: failure times, wait times, service times, etc. [Read more…] about Gamma Distribution: Uses, Parameters & Examples
The exponential distribution is a right-skewed continuous probability distribution that models variables in which small values occur more frequently than higher values. It is a unimodal distribution where small values have relatively high probabilities, which consistently decline as data values increase. Statisticians use the exponential distribution to model the amount of change in people’s pockets, the length of phone calls, and sales totals for customers. In all these cases, small values are more likely than larger values. [Read more…] about Exponential Distribution: Uses, Parameters & Examples
The Weibull distribution is a continuous probability distribution that can fit an extensive range of distribution shapes. Like the normal distribution, the Weibull distribution is unimodal and describes probabilities associated with continuous data. However, unlike the normal distribution, it can also model skewed data. In fact, its extreme flexibility allows it to model both left- and right-skewed data. [Read more…] about Weibull Distribution: Uses, Parameters & Examples
The Poisson distribution is a discrete probability distribution that describes probabilities for counts of events that occur in a specified observation space. It is named after Siméon Denis Poisson.
In statistics, count data represent the number of events or characteristics over a given length of time, area, volume, etc. For example, you can count the number of cigarettes smoked per day, meteors seen per hour, the number of defects in a batch, and the occurrence of a particular crime by county. [Read more…] about Poisson Distribution: Definition & Uses
Use dot plots to display the distribution of your sample data when you have continuous variables. These graphs stack dots along the horizontal X-axis to represent the frequencies of different values. More dots indicate greater frequency. Each dot represents a set number of observations. [Read more…] about Dot Plots: Using, Examples, and Interpreting