Which measure of variability is the average of the absolute values of the difference between the individual data and the derived mean?
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Variability describes how far apart data points lie from each other and from the center of a distribution. Along with measures of central tendency, measures of variability give you descriptive statistics that summarize your data. Show
Variability is also referred to as spread, scatter or dispersion. It is most commonly measured with the following:
Why does variability matter?While the central tendency, or average, tells you where most of your points lie, variability summarizes how far apart they are. This is important because the amount of variability determines how well you can generalize results from the sample to your population. Low variability is ideal because it means that you can better predict information about the population based on sample data. High variability means that the values are less consistent, so it’s harder to make predictions. Data sets can have the same central tendency but different levels of variability or vice versa. If you know only the central tendency or the variability, you can’t say anything about the other aspect. Both of them together give you a complete picture of your data. Example: Variability in normal distributionsYou are investigating the amounts of time spent on phones daily by different groups of people.Using simple random samples, you collect data from 3 groups:
All three of your samples have the same average phone use, at 195 minutes or 3 hours and 15 minutes. This is the x-axis value where the peak of the curves are. Although the data follows a normal distribution, each sample has different spreads. Sample A has the largest variability while Sample C has the smallest variability. RangeThe range tells you the spread of your data from the lowest to the highest value in the distribution. It’s the easiest measure of variability to calculate. To find the range, simply subtract the lowest value from the highest value in the data set. Range exampleYou have 8 data points from Sample A.
The highest value (H) is 324 and the lowest (L) is 72. R = H – L R = 324 – 72 = 252 The range of your data is 252 minutes. Because only 2 numbers are used, the range is influenced by outliers and doesn’t give you any information about the distribution of values. It’s best used in combination with other measures. What can proofreading do for your paper?Scribbr editors not only correct grammar and spelling mistakes, but also strengthen your writing by making sure your paper is free of vague language, redundant words and awkward phrasing.  See editing example Interquartile rangeThe interquartile range gives you the spread of the middle of your distribution. For any distribution that’s ordered from low to high, the interquartile range contains half of the values. While the first quartile (Q1) contains the first 25% of values, the fourth quartile (Q4) contains the last 25% of values. The interquartile range is the third quartile (Q3) minus the first quartile (Q1). This gives us the range of the middle half of a data set. Interquartile range exampleTo find the interquartile range of your 8 data points, you first find the values at Q1 and Q3.Multiply the number of values in the data set (8) by 0.25 for the 25th percentile (Q1) and by 0.75 for the 75th percentile (Q3). Q1 position: 0.25 x 8 = 2 Q3 position: 0.75 x 8 = 6 Q1 is the value in the 2nd position, which is 110. Q3 is the value in the 6th position, which is 287. IQR = Q3 – Q1 IQR = 287 – 110 = 177 The interquartile range of your data is 177 minutes. Just like the range, the interquartile range uses only 2 values in its calculation. But the IQR is less affected by outliers: the 2 values come from the middle half of the data set, so they are unlikely to be extreme scores. The IQR gives a consistent measure of variability for skewed as well as normal distributions. Five-number summaryEvery distribution can be organized using a five-number summary:
These five-number summaries can be easily visualized using box and whisker plots. Box and whisker plot exampleFor each of our samples, the horizontal lines in a box show Q1, the median and Q3, while the whiskers at the end show the highest and lowest values.Standard deviationThe standard deviation is the average amount of variability in your dataset. It tells you, on average, how far each score lies from the mean. The larger the standard deviation, the more variable the data set is. There are six steps for finding the standard deviation by hand:
Standard deviation example
Standard deviation exampleBecause you’re dealing with a sample, you use n – 1. n – 1 = 7 63904 / 7 = 9129.14 Standard deviation example s = √9129.14 = 95.54 The standard deviation of your data is 95.54. This means that on average, each score deviates from the mean by 95.54 points. Standard deviation formula for populationsIf you have data from the entire population, use the population standard deviation formula:
Standard deviation formula for samplesIf you have data from a sample, use the sample standard deviation formula:
Why use n – 1 for sample standard deviation?Samples are used to make statistical inferences about the population that they came from. When you have population data, you can get an exact value for population standard deviation. Since you collect data from every population member, the standard deviation reflects the precise amount of variability in your distribution, the population. But when you use sample data, your sample standard deviation is always used as an estimate of the population standard deviation. Using n in this formula tends to give you a biased estimate that consistently underestimates variability. Reducing the sample n to n – 1 makes the standard deviation artificially large, giving you a conservative estimate of variability. While this is not an unbiased estimate, it is a less biased estimate of standard deviation: it is better to overestimate rather than underestimate variability in samples. The difference between biased and conservative estimates of standard deviation gets much smaller when you have a large sample size. VarianceThe variance is the average of squared deviations from the mean. A deviation from the mean is how far a score lies from the mean. Variance is the square of the standard deviation. This means that the units of variance are much larger than those of a typical value of a data set. While it’s harder to interpret the variance number intuitively, it’s important to calculate variance for comparing different data sets in statistical tests like ANOVAs. Variance reflects the degree of spread in the data set. The more spread the data, the larger the variance is in relation to the mean. Variance exampleTo get variance, square the standard deviation.s = 95.5 s2 = 95.5 x 95.5 = 9129.14 The variance of your data is 9129.14. To find the variance by hand, perform all of the steps for standard deviation except for the final step. Variance formula for populations
Variance formula for samples
Biased versus unbiased estimates of varianceAn unbiased estimate in statistics is one that doesn’t consistently give you either high values or low values – it has no systematic bias. Just like for standard deviation, there are different formulas for population and sample variance. But while there is no unbiased estimate for standard deviation, there is one for sample variance. If the sample variance formula used the sample n, the sample variance would be biased towards lower numbers than expected. Reducing the sample n to n – 1 makes the variance artificially larger. In this case, bias is not only lowered but totally removed. The sample variance formula gives completely unbiased estimates of variance. So why isn’t the sample standard deviation also an unbiased estimate? That’s because sample standard deviation comes from finding the square root of sample variance. Since a square root isn’t a linear operation, like addition or subtraction, the unbiasedness of the sample variance formula isn’t carried over the sample standard deviation formula. What’s the best measure of variability?The best measure of variability depends on your level of measurement and distribution. Level of measurementFor data measured at an ordinal level, the range and interquartile range are the only appropriate measures of variability. For more complex interval and ratio levels, the standard deviation and variance are also applicable. DistributionFor normal distributions, all measures can be used. The standard deviation and variance are preferred because they take your whole data set into account, but this also means that they are easily influenced by outliers. For skewed distributions or data sets with outliers, the interquartile range is the best measure. It’s least affected by extreme values because it focuses on the spread in the middle of the data set. Frequently asked questions about variabilityWhat is variability? Variability tells you how far apart points lie from each other and from the center of a distribution or a data set. Variability is also referred to as spread, scatter or dispersion. Which measure of variability is a measure of the average distance from the mean?The standard deviation is the most commonly used measure for variability. This measure is related to the distance between the observations and the mean.
What is the absolute measure of variability?The absolute deviation, variance and standard deviation are such measures. The absolute and mean absolute deviation show the amount of deviation (variation) that occurs around the mean score. To find the total variability in our group of data, we simply add up the deviation of each score from the mean.
What measure of variability is the average of the squared difference of the values from the mean?The variance is a measure of variability. It is calculated by taking the average of squared deviations from the mean.
Which is the average of the sum of absolute difference of each measure from the mean?Gini's mean difference is the average absolute difference between any two different observations. Besides being robust and easy to interpret it happens to be 0.98 as efficient as SD if the distribution were actually Gaussian.
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