What term describe the amount of spread dispersion or variability of the item in a distribution?

Journal List
J Pharmacol Pharmacother
v.2(4); Oct-Dec 2011
PMC3198538

J Pharmacol Pharmacother. 2011 Oct-Dec; 2(4): 315–316.

INTRODUCTION

The measures of central tendency are not adequate to describe data. Two data sets can have the same mean but they can be entirely different. Thus to describe data, one needs to know the extent of variability. This is given by the measures of dispersion. Range, interquartile range, and standard deviation are the three commonly used measures of dispersion.

RANGE

The range is the difference between the largest and the smallest observation in the data. The prime advantage of this measure of dispersion is that it is easy to calculate. On the other hand, it has lot of disadvantages. It is very sensitive to outliers and does not use all the observations in a data set.[1] It is more informative to provide the minimum and the maximum values rather than providing the range.

INTERQUARTILE RANGE

Interquartile range is defined as the difference between the 25th and 75th percentile (also called the first and third quartile). Hence the interquartile range describes the middle 50% of observations. If the interquartile range is large it means that the middle 50% of observations are spaced wide apart. The important advantage of interquartile range is that it can be used as a measure of variability if the extreme values are not being recorded exactly (as in case of open-ended class intervals in the frequency distribution).[2] Other advantageous feature is that it is not affected by extreme values. The main disadvantage in using interquartile range as a measure of dispersion is that it is not amenable to mathematical manipulation.

STANDARD DEVIATION

Standard deviation (SD) is the most commonly used measure of dispersion. It is a measure of spread of data about the mean. SD is the square root of sum of squared deviation from the mean divided by the number of observations.

This formula is a definitional one and for calculations, an easier formula is used. The computational formula also avoids the rounding errors during calculation.

In both these formulas n - 1 is used instead of n in the denominator, as this produces a more accurate estimate of population SD.

The reason why SD is a very useful measure of dispersion is that, if the observations are from a normal distribution, then[3] 68% of observations lie between mean ± 1 SD 95% of observations lie between mean ± 2 SD and 99.7% of observations lie between mean ± 3 SD

The other advantage of SD is that along with mean it can be used to detect skewness. The disadvantage of SD is that it is an inappropriate measure of dispersion for skewed data.

APPROPRIATE USE OF MEASURES OF DISPERSION

SD is used as a measure of dispersion when mean is used as measure of central tendency (ie, for symmetric numerical data).

For ordinal data or skewed numerical data, median and interquartile range are used.[4]

Footnotes

Source of Support: Nil

Conflict of Interest: None declared.

REFERENCES

1. Swinscow TD, Campbell MJ. (Indian) 10th ed. New Delhi: Viva Books Private Limited; 2003. Statistics at square one. [Google Scholar]

2. Sundaram KR, Dwivedi SN, Sreenivas V. 1st ed. New Delhi: B.I Publications Pvt Ltd; 2010. Medical statistics principles and methods. [Google Scholar]

3. Gravetter FJ, Wallnau LB. 5th ed. Belmont: Wadsworth – Thomson Learning; 2000. Statistics for the behavioral sciences. [Google Scholar]

4. Dawson B, Trapp RG. 4th ed. New York: Mc-Graw Hill; 2004. Basic and Clinical Biostatistics. [Google Scholar]

Articles from Journal of Pharmacology & Pharmacotherapeutics are provided here courtesy of Wolters Kluwer -- Medknow Publications

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.

Variability is also referred to as spread, scatter or dispersion. It is most commonly measured with the following:

Range: the difference between the highest and lowest values
Interquartile range: the range of the middle half of a distribution
Standard deviation: average distance from the mean
Variance: average of squared distances from the mean

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:

Sample A: high school students,
Sample B: college students,
Sample C: adult full-time employees.

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.

Range

The 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.

Data (minutes)	72	110	134	190	238	287	305	324

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.

Interquartile range

The 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 summary

Every distribution can be organized using a five-number summary:

Lowest value
Q1: 25th percentile
Q2: the median
Q3: 75th percentile
Highest value (Q4)

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 deviation

The 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:

List each score and find their mean.
Subtract the mean from each score to get the deviation from the mean.
Square each of these deviations.
Add up all of the squared deviations.
Divide the sum of the squared deviations by n – 1 (for a sample) or N (for a population).
Find the square root of the number you found.

Standard deviation example

Step 1: Data (minutes)	Step 2: Deviation from mean	Steps 3 + 4: Squared deviation
72	72 – 207.5 = -135.5	18360.25
110	110 – 207.5 = -97.5	9506.25
134	134 – 207.5 = -73.5	5402.25
190	190 – 207.5 = -17.5	306.25
238	238 – 207.5 = 30.5	930.25
287	287 – 207.5 = 79.5	6320.25
305	305 – 207.5 = 97.5	9506.25
324	324 – 207.5 = 116.5	13572.25
Mean = 207.5	Sum = 0	Sum of squares = 63904

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 populations

If you have data from the entire population, use the population standard deviation formula:

Formula	Explanation
	= population standard deviation = sum of… = each value = population mean = number of values in the population

Standard deviation formula for samples

If you have data from a sample, use the sample standard deviation formula:

Formula	Explanation
	= sample standard deviation = sum of… = each value = sample mean = number of values in the sample

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.

Variance

The 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

Formula	Explanation
	= population variance = sum of… = each value = population mean = number of values in the population

Variance formula for samples

Formula	Explanation
	= sample variance = sum of… = each value = sample mean = number of values in the sample

Biased versus unbiased estimates of variance

An 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 measurement

For 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.

Distribution

For 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 variability

What 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.

Sources in this article

We strongly encourage students to use sources in their work. You can cite our article (APA Style) or take a deep dive into the articles below.

This Scribbr article

Bhandari, P. (September 2, 2022). Variability | Calculating Range, IQR, Variance, Standard Deviation. Scribbr. Retrieved October 13, 2022, from https://www.scribbr.com/statistics/variability/

What describes the amount of spread dispersion or variability of the items in a distribution?

Introduction to standard deviation Standard deviation measures the spread of a data distribution. The more spread out a data distribution is, the greater its standard deviation.

Which is used to describe the amount of spread dispersion?

The variance and the standard deviation are measures of the spread of the data around the mean. They summarise how close each observed data value is to the mean value.

What is the variability of the items in a distribution?

Variability is most commonly measured with the following descriptive statistics: Range: the difference between the highest and lowest values. Interquartile range: the range of the middle half of a distribution. Standard deviation: average distance from the mean.