Quartile Deviation: Measuring Data Spread Around the Median

When studying a dataset, knowing the average is rarely enough. Two datasets can have the same mean yet look completely different in terms of how spread out their values are. That is why measures of dispersion matter – and Quartile Deviation (QD), also known as the semi-interquartile range, is one of the most reliable tools for understanding how data is distributed around the median. It focuses specifically on the middle 50% of a dataset, giving a clear picture of central spread without being thrown off by extreme values at either end.

What is quartile deviation? 🔗

Quartile Deviation is a statistical measure of spread or dispersion. It tells us how much the central portion of a dataset varies around the median. More precisely, it measures the average distance from the median to the first and third quartiles. To understand it properly, we first need to understand what quartiles are.

Quartiles are values that divide a ranked dataset into four equal parts. There are three quartile points:

• Q1 (First Quartile) – the value below which 25% of the data falls (25th percentile).
• Q2 (Second Quartile) – the median, the midpoint of the data (50th percentile).
• Q3 (Third Quartile) – the value below which 75% of the data falls (75th percentile).

The gap between Q1 and Q3 is called the Interquartile Range (IQR). Quartile Deviation is simply half of this IQR, and it represents the spread of the middle 50% of the data.

The formula for quartile deviation 🔗

The formula is straightforward:

QD = (Q3 − Q1) / 2

Where Q1 is the first quartile (25th percentile) and Q3 is the third quartile (75th percentile). Since half the scores in a distribution lie between Q3 and Q1, the semi-interquartile range is half the distance needed to cover half the scores.

A smaller QD means the data points in the middle are packed closely together – low variability. A larger QD means the central data is more spread out – higher variability.

How to calculate quartile deviation: step by step 🔗

For ungrouped data 🔗

Ungrouped data is simply a raw list of values. Here is how to calculate QD from such data:

1. Arrange the data in ascending order. Always sort values from smallest to largest before locating quartiles.
2. Find Q1. For ungrouped data, use the formula Q1 = [(n+1)/4]th item, where n is the total number of observations.
3. Find Q3. Use the formula Q3 = [3(n+1)/4]th item.
4. Apply the QD formula. Subtract Q1 from Q3 and divide by 2.

Example: Consider this dataset of 10 scores: 12, 15, 18, 21, 22, 23, 25, 28, 30, 32.

• Q1 = median of the lower half (12, 15, 18, 21, 22) = 18
• Q3 = median of the upper half (23, 25, 28, 30, 32) = 28
• QD = (28 − 18) / 2 = 5

This tells us that the scores in the middle 50% of this dataset deviate by about 5 points from the median on average.

For grouped data 🔗

When data is arranged in class intervals with frequencies – known as grouped data – the process requires an additional step. Since individual values are not available, quartiles are estimated using an interpolation formula based on class boundaries, cumulative frequencies, and class widths.

The general formula for finding quartiles in grouped data is:

Qn = l + [(n·N/4 − C) / f] × h

Where:

• l = lower boundary of the quartile class
• N = total number of observations
• C = cumulative frequency of the class before the quartile class
• f = frequency of the quartile class
• h = class width

Once Q1 and Q3 are estimated using this formula, the same QD formula – (Q3 − Q1) / 2 – is applied. The underlying concept remains the same whether data is grouped or ungrouped; only the method for locating quartiles differs.

The coefficient of quartile deviation 🔗

QD is an absolute measure of dispersion – it is expressed in the same units as the data. When you want to compare the spread across two different datasets (especially when the units or scales differ), you need a relative measure. That is where the Coefficient of Quartile Deviation comes in.

Coefficient of QD = (Q3 − Q1) / (Q3 + Q1)

This relative measure is unit-free, making it suitable for comparing the dispersion of datasets with different scales or units. A higher coefficient means greater relative spread in the central data; a lower coefficient indicates more consistency.

Why quartile deviation is resistant to outliers 🔗

One of the biggest practical advantages of QD is its robustness. Because it is based on values from the middle half of the distribution, it is unlikely to be influenced by outliers.

Consider how standard deviation works: it uses every data point in the dataset, including extreme values at both ends. A single unusually large or small score can inflate the standard deviation considerably. When an outlier is added to a dataset, the IQR – and therefore the QD – barely changes, while the standard deviation can increase dramatically.

This makes QD particularly valuable in real-world psychological and educational research, where datasets often contain scores that deviate sharply from the norm – whether due to extreme performance, measurement error, or population heterogeneity.

When should you use quartile deviation? 🔗

QD is not always the best measure of spread – it depends on the nature of your data and what you want to communicate. Here is when QD is the right choice:

• When the median is your preferred measure of central tendency. QD and the median are natural companions. When data is skewed or contains outliers, neither the mean nor the standard deviation describe the data well – the median and QD together give a more honest picture.
• When the data is skewed. For skewed distributions or datasets with outliers, the interquartile range – and by extension, QD – is the best measure, as it focuses on the central portion of the data and is least affected by extreme values.
• When you are working with ordinal data. QD can be meaningfully calculated for ordinal scales, where mean and standard deviation may not be appropriate.
• When you want a quick summary of central spread. QD provides a focused picture of variability around the median without the computational complexity of standard deviation.

Conversely, if your data is normally distributed and free of extreme outliers, standard deviation is typically preferable because it uses every value in the dataset and gives a more sensitive measure of overall variability.

Advantages and limitations of quartile deviation 🔗

Advantages 🔗

• Resistant to extreme values: QD is less sensitive to extreme scores than range, variance, or standard deviation, making it more reliable when outliers are present.
• Works for both grouped and ungrouped data: The same logic applies to raw scores and frequency distributions alike.
• Easy to interpret: QD gives a concrete, intuitive sense of how closely bunched the central data is around the median.
• Useful with box plots: QD corresponds to half the length of the box in a box-and-whisker plot, making it a direct visual aid in data exploration.

Limitations 🔗

• Ignores the tails: By focusing only on the middle 50%, QD gives no information about the spread of the bottom 25% or top 25% of the data.
• Does not reflect the full distribution shape: QD cannot distinguish between symmetric and skewed distributions – two very different datasets can produce the same QD.
• More subject to sampling fluctuation in normal distributions: In approximately normal distributions, QD is more susceptible to sampling variability than standard deviation, which is why it is not the preferred measure when data is roughly normal.

Quartile deviation and the median: a natural pair 🔗

There is an elegant symmetry between QD and the median. In a symmetric distribution, an interval stretching one QD below and one QD above the median will contain approximately half of all scores. This makes QD a practical companion to the median – together they efficiently summarize both the center and the typical spread of a dataset.

In psychological research, this pairing is especially common when reporting performance data on tests and assessments, income distributions, reaction times, or any measure that tends to be skewed. For instance, if a researcher reports the median response time in a cognitive experiment, they would pair it with QD rather than standard deviation to avoid distortion from a few unusually slow or fast responses.

According to the APA Dictionary of Psychology, quartile deviation is formally defined as the value that is halfway between the first and third quartiles – essentially half the interquartile range – reflecting its role as a balanced measure of central spread.

A quick summary 🔗

Quartile Deviation is calculated as QD = (Q3 − Q1) / 2. It measures the spread of the middle 50% of data around the median. It is robust against outliers, applicable to both grouped and ungrouped data, and most appropriate when the median is the preferred measure of central tendency or when the data is skewed. Its relative counterpart – the coefficient of QD – allows meaningful comparison across datasets with different scales. While it does not capture the full picture of a distribution, within its defined scope, it is one of the most dependable and straightforward measures of dispersion available.

What do you think? If two students score the same median on a psychology exam but have very different quartile deviations, what does that tell you about the consistency of each student’s performance? And in what kinds of psychological research scenarios do you think QD would be more useful than standard deviation?

References

1. https://www.statisticshowto.com/semi-interquartile-range-quartile-deviation/
2. https://en.wikipedia.org/wiki/Interquartile_range
3. https://formalpsychology.com/quartile-deviation-q-d-measurement-of-variability/
4. https://davidmlane.com/hyperstat/A48607.html
5. https://testbook.com/maths/quartile-deviation
6. https://statistics.arabpsychology.com/calculate-quartiles-for-grouped-data/
7. https://itfeature.com/statistics/dispersion/quartile-deviation/
8. https://www.cuemath.com/data/quartile-deviation/
9. https://www.scribbr.com/frequently-asked-questions/when-should-i-use-the-interquartile-range/
10. https://www.statology.org/interquartile-range-vs-standard-deviation/
11. https://courses.lumenlearning.com/wm-concepts-statistics/chapter/standard-deviation-4-of-4/
12. https://www.scribbr.com/statistics/variability/
13. https://www.quanthub.com/how-do-outliers-impact-measures-of-variability/
14. https://dictionary.apa.org/quartile-deviation