Range

Hand a statistician a list of numbers and the very first question they ask is often the crudest one: how far apart are the biggest and the smallest values? That single subtraction is the range, the simplest of all measures of dispersion. It needs no averaging, no squaring, no sorting beyond finding two extremes — just xmax⁡−xmin⁡x_{\max} - x_{\min}. In CBSE Class 11 Statistics, range is the opening act before the more careful measures — mean deviation, variance, standard deviation — that use every observation rather than just the two outermost ones. This article keeps the Pass-1 scope intact: raw-data range, grouped-data range, and the coefficient of range, with both worked temperature and marks examples preserved exactly, now with a fuller look at why range behaves the way it does and where it quietly misleads.

The core insight

Range is the difference between the largest and smallest values in a data set, Range=xmax⁡−xmin⁡\text{Range} = x_{\max} - x_{\min} (or, for grouped data, the upper limit of the last class minus the lower limit of the first) — the fastest possible measure of spread, but also the crudest, since it uses only two data points and ignores everything in between.

The core idea

For raw (ungrouped) data, range is defined directly from the two extreme observations:

Range=xmax⁡−xmin⁡\text{Range} = x_{\max} - x_{\min}

where: xmax⁡x_{\max} is the largest value in the data set and xmin⁡x_{\min} is the smallest. Both must come from the actual list of numbers — nothing is computed from an average. For a grouped frequency distribution, individual maximum and minimum values are not visible — the data has already been bucketed into class intervals. So range is redefined using the boundaries of the outermost classes:

Range=(upper limit of last class)−(lower limit of first class)\text{Range} = (\text{upper limit of last class}) - (\text{lower limit of first class})

where: the "last class" is the highest interval in the table and the "first class" is the lowest; their limits, not any single data point, define the spread. A companion quantity, the coefficient of range, converts range into a unit-free ratio so that two data sets measured on different scales (say, marks out of 50 versus marks out of 100) can be compared fairly:

Coefficient of range=xmax⁡−xmin⁡xmax⁡+xmin⁡\text{Coefficient of range} = \frac{x_{\max} - x_{\min}}{x_{\max} + x_{\min}}

where: the numerator is the ordinary range and the denominator is the sum of the two extremes; the result is a pure number, typically reported to two or three decimal places. Why does range behave the way it does? Because it is built from only two numbers, it is extremely sensitive to outliers — a single freak observation can double or triple the range while everything else in the data stays clustered tightly together. Conversely, range can look deceptively small if the sample simply never captured an extreme value that exists in the wider population. This is the structural weakness that motivates every later measure in the chapter: mean deviation and standard deviation both fold in every observation, so one wild point cannot dominate the whole picture the way it can with range.

Figure
A number line showing ten scattered points with the minimum and maximum marked and a bracket spanning the range between them
Range spans only the two extreme points x min x max Range = x max minus x min The eight blue points between the extremes have no effect on the range at all
Derivation ARange and the coefficient of range from raw data

The goal of this derivation is to show precisely how the two range-related quantities are built from a single sorted or scanned pass over a raw data set, so that a student can compute both without confusing which one uses a difference and which one uses a ratio, and can see why the coefficient of range is bounded between 0 and 1 for any positive-valued data set.

Step 1. Scan the data set once and record the largest value seen, xmax⁡x_{\max}, and the smallest value seen, xmin⁡x_{\min}. No sorting is strictly required — a single pass comparing each new value against the running maximum and minimum suffices, though sorting makes the extremes obvious at a glance.

Step 2. Subtract to get the range. Range=xmax⁡−xmin⁡\text{Range} = x_{\max} - x_{\min}. This is an absolute measure — it carries the same units as the original data (degrees, marks, rupees, whatever was measured) and grows or shrinks with the scale of the data.

Step 3. Divide by the sum of the extremes to get the coefficient of range. Coefficient of range=xmax⁡−xmin⁡xmax⁡+xmin⁡\text{Coefficient of range} = \dfrac{x_{\max}-x_{\min}}{x_{\max}+x_{\min}}. Because the numerator is always less than the denominator whenever xmin⁡>0x_{\min} > 0, this ratio always lies strictly between 0 and 1 for positive data — a coefficient near 0 signals the extremes are close together relative to their size, while a coefficient near 1 signals one extreme is close to zero and the other is comparatively huge.

Step 4. Interpret. The plain range answers "how many units apart are the extremes," in the original units of the data. The coefficient of range answers "how large is that gap relative to the overall scale," a dimensionless number that lets two data sets with different units or magnitudes be compared side by side.

Worked example

The daily maximum temperatures (in degrees C) recorded in a city over 10 days are:

28, 30, 27, 31, 29, 33, 26, 32, 30, 2928,\ 30,\ 27,\ 31,\ 29,\ 33,\ 26,\ 32,\ 30,\ 29

Step 1. Identify the maximum and minimum. Scanning the list: xmax⁡=33x_{\max} = 33, xmin⁡=26x_{\min} = 26.

Step 2. Apply the range formula. Range=xmax⁡−xmin⁡=33−26=7\text{Range} = x_{\max} - x_{\min} = 33 - 26 = 7 degrees C.

Step 3. Compute the coefficient of range as an optional relative check. Coefficient of range=33−2633+26=759≈0.119\text{Coefficient of range} = \dfrac{33-26}{33+26} = \dfrac{7}{59} \approx 0.119.

Step 4. Interpret. The temperature varied by 7 degrees C over the ten days, and a coefficient of about 0.119 is small, consistent with a fairly stable pattern rather than wild swings.

Competitive stretch

Now take the grouped frequency distribution of marks obtained by 50 students in a test: class intervals 00–1010, 1010–2020, 2020–3030, 3030–4040, 4040–5050 with frequencies 4,10,20,12,44, 10, 20, 12, 4. The lowest class is 00–1010, giving a lower limit of 00; the highest class is 4040–5050, giving an upper limit of 5050. So Range=50−0=50 marks\text{Range} = 50 - 0 = \textbf{50 marks}. Notice that 32 of the 50 students (the middle two intervals) scored between 20 and 40 — tightly bunched — yet the range of 50 says nothing about that concentration, because it only sees the two outer boundaries. This is exactly why range is called the weakest of the four dispersion measures studied in this chapter.

Common traps

  • For grouped data, using the class limits (0 and 50 above) rather than trying to find an individual "maximum" or "minimum" value, which simply isn't available once data has been grouped.
  • Assuming a small range always means "consistent" data — a small sample can miss extreme values that exist in the wider population, making the range misleadingly small.
  • Confusing range (one number, xmax⁡−xmin⁡x_{\max}-x_{\min}) with class width — related but not the same concept.
  • Treating the coefficient of range as compulsory for every problem — it is only needed when comparing two data sets on different scales.

Ask the tutor: Want to practise computing the range for a grouped distribution or check whether an outlier is distorting a data set's spread — ask the AI tutor to work through a fresh example with you.

Why is range considered a weak measure of dispersion if it is so easy to calculate

Easy to calculate does not mean informative. Range uses only two values out of the entire data set, so it completely ignores how the remaining values are distributed. Two data sets can share the exact same range while one is tightly clustered and the other spread evenly across the interval. Measures like mean deviation and standard deviation use every observation, so they capture the shape of the spread, not just its outer boundary — that is why range is usually just a first, rough check.

Can the range of grouped data ever exceed the range you would get from the original raw data before grouping

Yes, slightly. Grouping uses the class limits (say 0 and 50) rather than the actual smallest and largest observed values (which might be 3 and 47). Since class limits sit at or beyond the true extremes, the grouped range is always greater than or equal to the raw-data range. This is a small distortion introduced by the act of grouping itself, and it is worth remembering when comparing a grouped-data range to a raw-data range for the same underlying data.

When would I actually use the coefficient of range instead of the plain range

Use it when comparing spread across two data sets that are not on the same scale — for instance, marks out of 50 in one class versus marks out of 100 in another, or rainfall in millimetres versus temperature in degrees. The plain range would be misleading because the units and magnitudes differ, but the coefficient of range is a unit-free ratio between 0 and 1, so it lets you say which data set is relatively more spread out regardless of the original scale.

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Where this leads