Range, Variance & Standard Deviation | Definition and Calculation

Aamir Fidai, Devin Kowalczyk
  • Author
    Aamir Fidai

    Dr. Aamir Fidai has taught Algebra 2, Precalculus, and Calculus to high school students for over 10 years. Dr. Fidai has a Ph.D. in Curriculum and Instruction from Texas A&M University where he also taught Mathematics Education courses to pre-service elementary school teachers.

  • Instructor
    Devin Kowalczyk

    Devin has taught psychology and has a master's degree in clinical forensic psychology. He is working on his PhD.

Learn the meaning of measures of variability in statistics. Study the four measures of variability and their formulas: range, variance, and standard deviation. Updated: 03/12/2022

What Are Measures of Variability?


Study of variation or measures of variability is one of the most important aspects of statistics and data analysis. Analysis of variation allows researchers and decision makers to make informed decisions in real-world situations. For example, a manufacturing company is looking to buy some ropes and is looking at two different suppliers. The manufacturer would like the strength of those ropes to be at least 50 pounds on average. Both suppliers claim the strength of their ropes is on average 50 pounds. How would the manufacturer decide which supplier to chose only knowing the mean strength of the ropes from each supplier? This is a perfect situation where information about the variation of the strength of ropes from two suppliers would be useful in making a decision.

Variation describes the spread of the data set or how scattered the dataset is. In other words, the measure of variation tells researchers and decision makers how far or close each data point is from the mean in a given data set. So, what is the measure of variation? The four most powerful and commonly used methods for calculating measures of variations are range, interquartile range, variance, and standard deviation.


Variance and Standard Deviation

Variance in statistics refers to how widely the data is scattered within a dataset or the vertical spread of the dataset. The variance (V) of a data set can be calculated in a few simple steps using the formula below. The spread or the scatter of the dataset refers to the distance of each data point from the average or mean value of the data set.


$$V = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2 $$


Standard deviation is the square root of the variance and is in the same units of measurement as the dataset itself. For example, a researcher wants to analyze a dataset containing daily stock prices for a tech company. The stock prices listed in U.S dollars. When the researcher calculates the standard deviation of the data set to determine the variation with the data, this measure of dispersion is also in U.S dollars. This makes explaining the amount of variation within the data set easy for the researcher and easy to understand for their audience. Standard deviation of a dataset helps stakeholders make decision in real life situation in all walks of life including medicine, industry, and academia, From deciding on which fuel type to use and which delivery route to take in the transportation industry to the efficacy of a vaccine in fighting a pandemic, standard deviation is an important tool which is used to make real-world decisions. The standard deviation is a measure of variability and is more often used to identify any given data value with the first, second, or third standard deviation on a normal distribution curve. Standard deviation (SD) can be calculated using the formula below.


$$SD = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2} $$


Both variance and standard deviation measure the spread of data about the mean of the dataset. Standard deviation is the preferred method for reporting variation within a dataset because standard deviation has the same units of measurements as the data set. A dataset is considered more reliable if the values for variance and standard deviation are small.


Range

Range is another commonly used measure of variation. However, range is less reliable and provides very little information about the variation in the data set. One advantage of reporting range is that the calculation of range is very easy. The range of a dataset can be calculated using the following formula.


$$Range = Max - Min $$

To calculate the range of a dataset simply subtract the minimum value in the dataset from the maximum value in the dataset. The measure of the range is also reported in the same units of measurement as the dataset itself.


Measures of Variability

I was going to write this about intelligence and intelligence quotients, but that got really complicated really fast. So, let's talk about obesity instead, because you're more likely to hear about the rising rates of obesity rather than the rising IQs. But this lesson is about weight and understanding the descriptions of it.

Weight, like so many other things, is not static or unchanging. Not everyone who is 6 feet tall is 200 pounds - there is some variability. When reporting these numbers or reviewing them for a project, a researcher needs to understand how much difference there is in the scores. This is where we will look at measures of variability, which are statistical procedures to describe how spread out the data is. They are:

  • Range: defined as a single number representing the spread of the data
  • Standard deviation: defined as a number representing how far from the average each score is
  • Variance: defined as a number indicating how spread out the data is

When trying to understand how spread out the data is, we, as researchers, need to differentiate and know the difference between population and sample. A population is defined as the complete collection to be studied, like all the police officers in your city. A sample is defined as a section of the population and would be a selection of police officers you are studying. This can be anywhere from 1% to 99% of them.

When researchers do psychological experiments, they often must work with samples, because to find everyone in the population is nearly impossible. If you want a population data set, such as the world's weight, for example, that would be about seven billion data points. If you wanted the population data set of everyone in California, then that means you need about 33 million data points. In my own town, this is about 100,000 people. The trick is trying to make your sample data look like the population, which means you need to find measures on how variable your data is compared to the estimated population.

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Range

Let's go back to our study on obesity. What's the range of weights we'll be looking at? Range is simply taking the highest score and subtracting the lowest score from it. It's pretty simple to find. If the heaviest person is 800 pounds and the smallest person is 100, then our range is 700 pounds. Remember: to do range, you will need to have scores that have some variability. For example, weight has a large variability in the scores and has a meaningful range. A five-question quiz would not have a very meaningful range because the largest possible range is five.

Range has a simple and easy to understand purpose as well: to quickly and easily inform us on how wide the scores are. If we're doing a study and using a sample, we need to know how representative of the population our sample is. For example, if we are looking at weight and depression and our range is 50 pounds, then we don't have a very wide range, and it's not representative of the population. This may limit the findings on how depression affects weight because we're only looking at either the lowest recorded weights or the overweight instead of comparing the two. If our range is 500 pounds, now we're looking at a broader sample and a likely more representative sample of weight and how it affects depression.

Standard Deviation

While range is about how much your data covers, standard deviation has to do more with how much difference there is between the scores. If all of the scores are grouped around the average, then your standard deviation will be lower. If your scores are all over the map and not grouped together at all, then your standard deviation will be huge. The steps for calculating it are:

  • Calculate the average
  • Calculate the deviations, which are the scores minus the average
  • Square the deviations
  • Sum up the squared deviations
  • Divide this by the number of scores in your data set (or multiply by 1/N, same thing)
  • Take the square root

Formula for finding standard deviation
formula for standard deviation

The standard deviation can also be found in Excel using the STDDEV commands for a data set.

The formula takes advantage of statistical language and is not as complicated as it seems. The part in the parentheses above is the first two steps, subtracting the average (the x with the line over it) and the score (represented by xi). Then you square each result. The big, funny E (called sigma) means that you add up all the squared deviations. Then you multiply the sum by one divided by the number of scores in your sample. The last step is square rooting to get your standard deviation, which is represented on the left side of the equation by the Sn.

If you have a group of scores and they're all clustered around the mean, then our second step of calculating the squared deviations would result in a smaller number. This would make all the math later much smaller, and thus our standard deviation smaller.

When all our scores are clustered around the middle, it would look like the graph below, with all the scores making a huge bump in the middle.

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Video Transcript

Measures of Variability

I was going to write this about intelligence and intelligence quotients, but that got really complicated really fast. So, let's talk about obesity instead, because you're more likely to hear about the rising rates of obesity rather than the rising IQs. But this lesson is about weight and understanding the descriptions of it.

Weight, like so many other things, is not static or unchanging. Not everyone who is 6 feet tall is 200 pounds - there is some variability. When reporting these numbers or reviewing them for a project, a researcher needs to understand how much difference there is in the scores. This is where we will look at measures of variability, which are statistical procedures to describe how spread out the data is. They are:

  • Range: defined as a single number representing the spread of the data
  • Standard deviation: defined as a number representing how far from the average each score is
  • Variance: defined as a number indicating how spread out the data is

When trying to understand how spread out the data is, we, as researchers, need to differentiate and know the difference between population and sample. A population is defined as the complete collection to be studied, like all the police officers in your city. A sample is defined as a section of the population and would be a selection of police officers you are studying. This can be anywhere from 1% to 99% of them.

When researchers do psychological experiments, they often must work with samples, because to find everyone in the population is nearly impossible. If you want a population data set, such as the world's weight, for example, that would be about seven billion data points. If you wanted the population data set of everyone in California, then that means you need about 33 million data points. In my own town, this is about 100,000 people. The trick is trying to make your sample data look like the population, which means you need to find measures on how variable your data is compared to the estimated population.

Range

Let's go back to our study on obesity. What's the range of weights we'll be looking at? Range is simply taking the highest score and subtracting the lowest score from it. It's pretty simple to find. If the heaviest person is 800 pounds and the smallest person is 100, then our range is 700 pounds. Remember: to do range, you will need to have scores that have some variability. For example, weight has a large variability in the scores and has a meaningful range. A five-question quiz would not have a very meaningful range because the largest possible range is five.

Range has a simple and easy to understand purpose as well: to quickly and easily inform us on how wide the scores are. If we're doing a study and using a sample, we need to know how representative of the population our sample is. For example, if we are looking at weight and depression and our range is 50 pounds, then we don't have a very wide range, and it's not representative of the population. This may limit the findings on how depression affects weight because we're only looking at either the lowest recorded weights or the overweight instead of comparing the two. If our range is 500 pounds, now we're looking at a broader sample and a likely more representative sample of weight and how it affects depression.

Standard Deviation

While range is about how much your data covers, standard deviation has to do more with how much difference there is between the scores. If all of the scores are grouped around the average, then your standard deviation will be lower. If your scores are all over the map and not grouped together at all, then your standard deviation will be huge. The steps for calculating it are:

  • Calculate the average
  • Calculate the deviations, which are the scores minus the average
  • Square the deviations
  • Sum up the squared deviations
  • Divide this by the number of scores in your data set (or multiply by 1/N, same thing)
  • Take the square root

Formula for finding standard deviation
formula for standard deviation

The standard deviation can also be found in Excel using the STDDEV commands for a data set.

The formula takes advantage of statistical language and is not as complicated as it seems. The part in the parentheses above is the first two steps, subtracting the average (the x with the line over it) and the score (represented by xi). Then you square each result. The big, funny E (called sigma) means that you add up all the squared deviations. Then you multiply the sum by one divided by the number of scores in your sample. The last step is square rooting to get your standard deviation, which is represented on the left side of the equation by the Sn.

If you have a group of scores and they're all clustered around the mean, then our second step of calculating the squared deviations would result in a smaller number. This would make all the math later much smaller, and thus our standard deviation smaller.

When all our scores are clustered around the middle, it would look like the graph below, with all the scores making a huge bump in the middle.

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Frequently Asked Questions

What does variability mean in statistics?

Variability in statistics refers to how scattered or spread out the data set is compared to the mean value of the dataset. The three most powerful and commonly used methods for calculating measures of variations are range, variance, and standard deviation.

What are the measures of variability?

The three most powerful and commonly used methods for calculating measures of variations are range, variance, and standard deviation. These measures of variation can inform us about how scattered or spread out the data set is compared to the mean value of the dataset.

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