Grouped Data Standard Deviation Calculator
Enter class midpoints and frequencies to get mean, standard deviation, variance, and CV instantly.
📊 What is a Grouped Data Standard Deviation Calculator?
Grouped data standard deviation is a measure of dispersion calculated from a frequency distribution table rather than raw individual values. When data is organized into class intervals (such as exam scores 20-30, 30-40, 40-50), only the midpoint and frequency of each class are known, not the exact values inside each class. The standard deviation formula is adapted to work with these midpoints and frequencies, producing a weighted estimate of the spread around the grouped mean.
This calculator is used wherever data is presented in a frequency table format. Common applications include analyzing exam score distributions (a class of 200 students grouped into score bands), income distributions in economics (households grouped by income range), quality control in manufacturing (product measurements grouped into tolerance bands), public health research (age-grouped disease incidence data), and market research surveys (Likert-scale responses grouped by category). Any situation where raw data has been summarized into class intervals and frequencies calls for the grouped data formula.
A key distinction is between population standard deviation (sigma, using N in the denominator) and sample standard deviation (s, using N minus 1). If the frequency table describes the entire population of interest, use sigma. If the table is a sample drawn from a larger population, use s. The sample formula applies Bessel's correction (dividing by N minus 1 instead of N) to remove bias from the variance estimate. For large N the difference is negligible, but for small samples (say, 10 to 30 total observations) the correction matters.
The grouped standard deviation is an approximation because it assumes all observations within a class equal the midpoint. This grouping error is unavoidable unless the raw data is available. The approximation improves as class width decreases and as the data is more uniformly distributed within each class. This calculator shows the full step-by-step working table including f times m, deviations from the mean, squared deviations, and weighted squared deviations, so you can verify every intermediate step.
📐 Formula
x̅ = ∑(f × m) ÷ ∑f
x̅ = weighted mean of the grouped data
f = frequency of each class
m = midpoint of each class interval
∑f = total frequency (N)
σ = √[ ∑f(m − x̅)² ÷ ∑f ]
σ = population standard deviation
s = √[ ∑f(m − x̅)² ÷ (∑f − 1) ] = sample standard deviation (Bessel's correction)
CV = (s ÷ |x̅|) × 100% = coefficient of variation
Example: Classes 20-30 (f=3), 30-40 (f=5), 40-50 (f=8) with midpoints 25, 35, 45: mean = (75+175+360)/16 = 38.125; variance = sum of f times (m - 38.125) squared, divided by 16 or 15.
📖 How to Use This Calculator
Steps
1
Enter class midpoints - Type the midpoint of each class interval in the Midpoint column. Midpoint = (lower + upper boundary) divided by 2. For class 20-30, midpoint = 25.
2
Enter class frequencies - Type the count of observations in each class in the Frequency column. All frequencies must be positive numbers.
3
Fill in all active classes - Fill rows from top to bottom. Leave unused rows blank. The calculator uses only rows where both midpoint and frequency are filled in.
4
Click Calculate - Press Calculate to instantly compute mean, population SD, sample SD, variance, and coefficient of variation.
5
Read the full working table - Scroll below the results to see the step-by-step table with all intermediate values for verification.
💡 Example Calculations
Example 1 - Exam scores (5 classes)
Test scores for 27 students grouped into 10-point bands
1
Classes and midpoints: 20-30 (m=25, f=3), 30-40 (m=35, f=5), 40-50 (m=45, f=8), 50-60 (m=55, f=7), 60-70 (m=65, f=4). Total N = 27.
2
Mean = (3×25 + 5×35 + 8×45 + 7×55 + 4×65) / 27 = (75 + 175 + 360 + 385 + 260) / 27 = 1255 / 27 = 46.481.
3
Compute f(m - x-bar) squared for each class, sum them = 3(25-46.481)² + 5(35-46.481)² + 8(45-46.481)² + 7(55-46.481)² + 4(65-46.481)² = 1384.26 + 660.93 + 17.57 + 508.14 + 1371.56 = 3942.46.
4
Population SD = square root of (3942.46 / 27) = square root of 146.02 = 12.084. Sample SD = square root of (3942.46 / 26) = 12.316.
Mean = 46.481 | Sample SD = 12.316 | Pop. SD = 12.084
Try this example →Example 2 - Heights (4 classes, cm)
Heights of 40 adults grouped into 5 cm bands
1
Classes: 155-160 (m=157.5, f=6), 160-165 (m=162.5, f=14), 165-170 (m=167.5, f=12), 170-175 (m=172.5, f=8). N = 40.
2
Mean = (6×157.5 + 14×162.5 + 12×167.5 + 8×172.5) / 40 = (945 + 2275 + 2010 + 1380) / 40 = 6610 / 40 = 165.25 cm.
3
Sample SD = square root of [sum f(m-165.25)² / 39]. Working: 6(157.5-165.25)² + 14(162.5-165.25)² + 12(167.5-165.25)² + 8(172.5-165.25)² = 361.5 + 106.75 + 60.75 + 420.5 = 949.5. SD = square root of (949.5 / 39) = 4.932 cm.
Mean = 165.25 cm | Sample SD = 4.932 cm | CV = 2.99%
Try this example →Example 3 - Monthly incomes (6 classes)
Monthly incomes of 60 households grouped into 1000-unit bands
1
Classes: 1000-2000 (m=1500, f=5), 2000-3000 (m=2500, f=12), 3000-4000 (m=3500, f=18), 4000-5000 (m=4500, f=14), 5000-6000 (m=5500, f=8), 6000-7000 (m=6500, f=3). N = 60.
2
Mean = (5×1500 + 12×2500 + 18×3500 + 14×4500 + 8×5500 + 3×6500) / 60 = (7500 + 30000 + 63000 + 63000 + 44000 + 19500) / 60 = 227000 / 60 = 3783.33.
3
Sample SD is computed from the sum of f(m - 3783.33) squared = 5(1500-3783.33)² + ... = 26,133,333.4 / 59 = 442,938.7. Sample SD = square root = 665.53 units.
Mean = 3783.33 | Sample SD = 1154.70 | CV = 30.5%
Try this example →❓ Frequently Asked Questions
How do you calculate standard deviation for grouped data?
Five steps: (1) Find the midpoint m of each class interval. (2) Compute the weighted mean: x-bar = sum of (f times m) divided by sum of f. (3) For each class, compute the squared deviation (m minus x-bar) squared. (4) Multiply by frequency: f times (m minus x-bar) squared. (5) Sum all these products, divide by N for population SD or by N minus 1 for sample SD, then take the square root. The detail table in this calculator shows every intermediate step.
What is the formula for grouped data standard deviation?
Population SD: sigma = square root of [sum of f(m minus x-bar) squared, divided by sum of f]. Sample SD: s = square root of [sum of f(m minus x-bar) squared, divided by (sum of f minus 1)]. Here f is the class frequency, m is the class midpoint, x-bar is the weighted mean, and the sum is over all classes. The sample formula uses Bessel's correction to give an unbiased estimate of population variance.
What is the midpoint of a class interval?
The midpoint is the average of the lower and upper class boundaries: midpoint = (lower + upper) divided by 2. For the class 20 to 30: midpoint = (20 + 30) / 2 = 25. For 30 to 40: midpoint = 35. For 160 to 165 cm: midpoint = 162.5 cm. The midpoint represents all values in the class when computing the grouped mean and standard deviation.
When should I use population SD versus sample SD for grouped data?
Use population SD (sigma) when your frequency table represents the entire population (all 200 students in one school). Use sample SD (s) when the table represents a sample from a larger population (200 students selected from all schools in a country). Sample SD uses N minus 1 in the denominator (Bessel's correction) to produce an unbiased estimate of population variance. For large N (above 50), the two values are nearly identical.
How accurate is grouped data standard deviation compared to raw data SD?
The grouped SD is an approximation because it treats all observations in a class as equal to the midpoint. The error depends on class width and how uniformly data is distributed within each class. For narrow, equal-width classes with roughly uniform data inside, the grouped SD closely matches the raw SD. For wide classes or skewed within-class distributions, the error can be several percent. The approximation has no systematic direction: it can overestimate or underestimate the raw SD.
What is the coefficient of variation in grouped data?
CV = (sample SD divided by mean) times 100%. It expresses relative variability as a percentage of the mean. A CV of 20% means the standard deviation is 20% of the mean. CV is useful for comparing two datasets with different units or scales: exam scores (mean 50, SD 10, CV = 20%) can be compared with heights (mean 165 cm, SD 5 cm, CV = 3%). The dataset with the higher CV has more relative variability.
Can I use this calculator for open-ended class intervals?
Open-ended classes (like "70 and above" or "below 10") have no natural midpoint, so they cannot be used directly in this calculator without assumption. A common approach is to estimate a midpoint based on context or the width of adjacent classes. For example, if the last closed class is 60-70 (width 10), you might assume the open class 70+ has midpoint 75. Any such assumption introduces additional approximation beyond normal grouping error.
What does a large standard deviation mean for grouped data?
A large standard deviation means the data is widely spread around the mean. Observations (or their midpoints) are far from the average on average. For grouped exam scores, a large SD means students scored very differently from each other, spanning a wide range of bands. A small SD means most students scored similarly, clustering around the mean band. The coefficient of variation (CV) scales SD relative to the mean for context.
How many class intervals should a frequency table have?
Most statistics guidelines recommend 5 to 15 classes. Sturges's rule: k = 1 + 3.322 times log base 10 of n (where n is the total frequency). For n = 50, k = 1 + 3.322 times 1.699 = 6.6, so about 6 to 7 classes. Too few classes lose distributional detail; too many create sparse frequencies that amplify variability. This calculator supports up to 10 classes.
What is the difference between grouped data mean and arithmetic mean?
The arithmetic mean uses individual data values: x-bar = sum of all x divided by n. The grouped data mean uses class midpoints weighted by frequencies: x-bar = sum of (f times m) divided by sum of f. If you have raw data, the arithmetic mean is exact. If you only have the frequency table, the grouped mean is an approximation that matches the arithmetic mean only when all values within each class happen to equal the midpoint.
Can I enter non-equal class widths in this calculator?
Yes. This calculator only needs the midpoint and frequency of each class. It does not require equal class widths. Simply enter the correct midpoint for each class regardless of width. However, note that unequal class widths affect the interpretation: a wider class with the same frequency has more uncertainty about where within the class the data falls, potentially increasing grouping error.
What is variance for grouped data and how does it relate to SD?
Population variance (sigma squared) = sum of f(m minus x-bar) squared, divided by sum of f. Sample variance (s squared) = the same numerator divided by (sum of f minus 1). Standard deviation is simply the square root of variance: sigma = square root of sigma squared, and s = square root of s squared. Variance is in squared units, making it hard to interpret directly. SD is in the same units as the original data, so it is the preferred measure for describing spread.