IQR (Interquartile Range)

The average survey completion time is 4 minutes. The standard deviation is 18 minutes. The minimum is 30 seconds, the maximum is 4 hours (someone left a tab open overnight).

The mean and SD are distorted by a few abnormal values. The median is 3 minutes, the IQR is 2 minutes. These two numbers paint an honest picture: the bulk of respondents finish within 2-4 minutes. The IQR is a measure of spread that does not react to outliers and shows where the real data lies.

Definition

IQR (Interquartile Range) is the difference between the third (Q3) and first (Q1) quartiles of a sample. It shows the range that contains the central 50% of observations. Unlike the standard deviation, the IQR is robust to outliers - extreme values do not affect its magnitude. It is used in descriptive statistics, for building box plots, and for the formal detection of outliers via the 1.5×IQR rule.

What quartiles are

Quartiles split ordered data into four equal parts:

• Q1 (25th percentile) - 25% of observations fall below this value
• Q2 (50th percentile) - this is the median, with 50% of observations below it
• Q3 (75th percentile) - 75% of observations fall below this value

IQR = Q3 - Q1. That is, the difference between the 25th and 75th percentiles. It is the range that captures the middle 50% of respondents - without the bottom 25% and without the top 25%.

Why the IQR is robust to outliers

Imagine a sample of survey completion times: 2, 3, 3, 4, 4, 5, 6, 7, 8, 240 minutes. One outlier - someone left a tab open for 4 hours.

• Mean: 28.2 minutes (distorted by a single value)
• Standard deviation: 74.7 (huge because of the outlier)
• Median: 4.5 minutes
• Q1 = 3, Q3 = 7, IQR = 4 minutes

The median and IQR tell the truth: a typical respondent finishes within 3-7 minutes. The mean and SD are uninformative. This is the main advantage of the IQR: it characterizes the bulk of the data rather than the extreme values.

The 1.5×IQR rule for finding outliers

A formal method for detecting outliers:

• Lower bound: Q1 - 1.5 × IQR
• Upper bound: Q3 + 1.5 × IQR
• Anything beyond these bounds is a potential outlier

For the time example: Q1 = 3, Q3 = 7, IQR = 4. Lower bound: 3 - 6 = -3 (negative, so there are no outliers on the low end). Upper: 7 + 6 = 13. The value of 240 minutes far exceeds 13 - formally an outlier.

The 1.5×IQR rule is the standard for box plots. A stricter version - 3×IQR - is used for "extreme" outliers. Both methods help automatically flag anomalies in survey data: suspiciously fast or slow completions, unrealistic answers to numeric questions (age of 150, income of a billion dollars).

Box plot and the IQR

A box plot (box-and-whisker plot) is a visualization in which the IQR plays a central role:

• The "box" - the bounds from Q1 to Q3 (the central 50%)
• The line inside the box - the median
• The "whiskers" - up to 1.5×IQR from the edges of the box
• Points beyond the whiskers - outliers

Comparing the distributions of two groups via box plots is a quick way to see the difference in the center (a shift in the median) and the spread (the width of the box) without computing any statistics. This is especially useful when analyzing survey data by segment: for example, comparing satisfaction scores across different regions in a single chart.

IQR vs standard deviation

The standard deviation and the IQR are both measures of spread, but they measure different things.

SD uses every value in its calculation - each deviation from the mean is squared. Outliers have a quadratic influence: a single value 10 standard deviations from the mean contributes 100 times more than a typical one.

IQR uses only two values - Q1 and Q3. Anything above Q3 or below Q1 does not affect the magnitude of the IQR. Adding an extreme outlier to the sample will not change the IQR.

When to use which:

• Data is close to a normal distribution and there are no outliers -> SD + mean
• The distribution is skewed or there are outliers -> median + IQR
• In reports for a general audience - the IQR is more intuitive (it falls within the "median 50%" range)

Example: IQR in analyzing satisfaction scores

Results of a CSAT survey on a 1-10 scale across two products:

Product A: 7, 7, 8, 8, 8, 9, 9, 9, 9, 10. Median = 8.5, Q1 = 8, Q3 = 9, IQR = 1.

Product B: 2, 4, 7, 7, 8, 8, 9, 9, 10, 10. Median = 8, Q1 = 7, Q3 = 9, IQR = 2.

The medians are close: 8.5 and 8. But the IQR reveals that in Product A the bulk of customers fall within the narrow 8-9 range - high uniformity. In Product B the range is wider: 7-9, and on top of that there are values of 2 and 4 (potential outliers by the 1.5×IQR rule). This means Product B has a segment of dissatisfied customers. Product A has no such segment - a consistently high rating. That is a direct, practical takeaway for the product team.

Common mistakes when working with the IQR

Automatically removing all outliers by the 1.5×IQR rule. This is a statistical boundary, not a diagnosis. Some "outliers" are real cases worth studying separately rather than discarding. This matters especially in small samples: removing 2-3 points out of 30 substantially changes the picture.

Using the IQR for small samples. When n < 10, the quartiles are estimated with large error, and the IQR becomes unstable. For very small samples it is better to use all the raw values or the range (max - min).

Comparing IQRs from different scales. An IQR of 2 on a 1-10 scale is a significant spread. An IQR of 2 on a 1-100 scale is tiny. For comparison, use a relative measure: IQR / median.

The IQR in survey data analysis

The IQR is applied for descriptive statistics of survey completion time, scale ratings, and quantitative answers (age, income, frequency of use). It is useful in segmentation: comparing the IQR across segments shows which groups give uniform answers and which vary widely.

For flagging suspicious responses and controlling data quality, the 1.5×IQR rule helps automatically find anomalies in completion time, numeric answers, and number of actions. Export your data via data export and compute the IQR in Excel using the QUARTILE.INC function.

The IQR is spread without the influence of extremes. When the mean and SD lie because of outliers, the median and IQR show the real picture. The 1.5×IQR rule gives a formal way to look for anomalies. For skewed distributions and data with outliers, this pair (median + IQR) is the standard choice in descriptive statistics.

Frequently asked questions

How does the IQR differ from the range?

The range is the difference between the maximum and the minimum. The IQR is the difference between Q3 and Q1. The range uses the extreme values and depends heavily on outliers. The IQR uses only the central quartiles and is robust to outliers. For real data the IQR is almost always more informative than the range.

How do you calculate quartiles in Excel?

The function QUARTILE.INC(range, 1) returns Q1, and QUARTILE.INC(range, 3) returns Q3. IQR = Q3 - Q1. Alternatively: PERCENTILE.INC(range, 0.25) and PERCENTILE.INC(range, 0.75).

Should values beyond the 1.5×IQR bounds always be removed?

No. The 1.5×IQR rule is a way to flag unusual observations, not an automatic deletion criterion. Before removing them you need to understand the nature of the outlier: is it a data-entry error, anomalous behavior (a bot, an inattentive respondent), or a genuine rare case? In the last case, deletion distorts the picture rather than improving it.

When is the IQR better than the standard deviation?

When the data is skewed or contains outliers. For normally distributed data without outliers, the SD is more informative - it uses all the values. For survey completion times, income, and number of actions (always right-skewed distributions with outliers), the IQR is more appropriate.

Can the IQR be zero?

Yes - when Q1 and Q3 coincide. This means at least half the answers have the same value. It occurs in binary or heavily skewed distributions - for example, if 60% of respondents chose "yes" and Q1 = Q3 = 1. In such cases the IQR is uninformative, and other description methods are needed (proportions, frequencies).