Grouped Data Standard Deviation Calculator

Enter class midpoints and frequencies to get mean, standard deviation, variance, and CV instantly.

Enter midpoint and frequency for each class interval (leave unused rows blank).

#	Midpoint (m)	Frequency (f)
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Sample SD (s)

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Population SD (σ)

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Mean (x̅)

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Sample Variance (s²)

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Pop. Variance (σ²)

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Coeff. of Variation

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Total Frequency (N)

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Classes Used

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📊 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

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.

Enter class frequencies - Type the count of observations in each class in the Frequency column. All frequencies must be positive numbers.

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.

Click Calculate - Press Calculate to instantly compute mean, population SD, sample SD, variance, and coefficient of variation.

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

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.

Mean = (3×25 + 5×35 + 8×45 + 7×55 + 4×65) / 27 = (75 + 175 + 360 + 385 + 260) / 27 = 1255 / 27 = 46.481.

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.

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

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Example 2 - Heights (4 classes, cm)

Heights of 40 adults grouped into 5 cm bands

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.

Mean = (6×157.5 + 14×162.5 + 12×167.5 + 8×172.5) / 40 = (945 + 2275 + 2010 + 1380) / 40 = 6610 / 40 = 165.25 cm.

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%

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Example 3 - Monthly incomes (6 classes)

Monthly incomes of 60 households grouped into 1000-unit bands

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.

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.

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%

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❓ 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.

🔗 Related Calculators

How do you calculate standard deviation for grouped data?

Standard deviation of grouped data is computed in five steps: (1) Find the midpoint m of each class interval. (2) Compute the weighted mean: x-bar = sum(f times m) divided by sum(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 sum(f) for population SD or by sum(f) minus 1 for sample SD, then take the square root.

What is the formula for grouped data standard deviation?

Population SD: sigma = square root of [sum of f times (m minus x-bar) squared, divided by sum of f]. Sample SD: s = square root of [sum of f times (m minus x-bar) squared, divided by (sum of f minus 1)]. Here f is the frequency of each class, m is the class midpoint, x-bar is the weighted mean, and the sum is over all classes.

What is the difference between grouped and ungrouped standard deviation?

Ungrouped (raw) standard deviation uses individual data values and is exact. Grouped standard deviation uses class midpoints to approximate the actual values, treating all observations in a class as if they equal the midpoint. This introduces grouping error. Grouped SD is an approximation: it matches the raw SD exactly only if all values in each class are indeed equal to the midpoint.

How do you find the midpoint of a class interval?

Midpoint = (lower class boundary + upper class boundary) divided by 2. For the class 20 to 30: midpoint = (20 + 30) / 2 = 25. For 30 to 40: midpoint = 35. For 40 to 50: midpoint = 45. Use these midpoints as the representative value for each class when computing the grouped mean and standard deviation.

What is the coefficient of variation in grouped data?

The coefficient of variation (CV) = (sample SD divided by mean) times 100, expressed as a percentage. It measures relative variability. For example, if the mean is 50 and the sample SD is 10, CV = 20%. CV is useful when comparing two datasets with different units or scales: a dataset with CV = 15% is less variable relative to its mean than one with CV = 30%, regardless of their absolute spreads.

When should I use population SD versus sample SD for grouped data?

Use population standard deviation (sigma) when the frequency table represents the entire population of interest: for example, the heights of all 200 students in a specific school. Use sample standard deviation (s) when the table represents a sample from a larger population: for example, 200 students selected from all schools in a city. Sample SD uses (sum of f) minus 1 in the denominator (Bessel's correction) to produce an unbiased estimate of the population variance.

What is the mean of grouped data formula?

Mean x-bar = sum(f times m) divided by sum(f), where f is the frequency of each class and m is the class midpoint. This is the weighted arithmetic mean, with frequencies serving as weights. For example: classes with midpoints 25, 35, 45 and frequencies 3, 5, 8 give mean = (3 times 25 + 5 times 35 + 8 times 45) / (3 + 5 + 8) = (75 + 175 + 360) / 16 = 610 / 16 = 38.125.

How many class intervals do I need for grouped data standard deviation?

You need at least 2 classes with positive frequencies. In practice, 5 to 15 classes typically balance accuracy and manageability. Sturges's rule suggests k = 1 + 3.322 times log base 10 of n (where n is the total frequency) as a starting point. Too few classes lose detail; too many classes create sparse frequencies and amplify grouping error. This calculator accepts up to 10 classes.

What is variance for grouped data?

Population variance (sigma squared) = sum of f times (m minus x-bar) squared, all divided by sum of f. Sample variance (s squared) = sum of f times (m minus x-bar) squared, all divided by (sum of f minus 1). Variance is the square of standard deviation. Standard deviation is preferred for interpretation because it is in the same units as the original data, while variance is in squared units.

How does grouped data SD compare to raw data SD for the same dataset?

When the same data is expressed as a frequency table, the grouped SD approximates the raw SD. The approximation improves as class width decreases and as the data within each class is more uniformly distributed. For broad classes (e.g., 20-unit wide intervals with non-uniform data inside), the grouped SD can differ from the raw SD by several percent. Using the midpoint assumption introduces bias if the data is skewed within classes.

Can I use this calculator for continuous and discrete grouped data?

Yes. For continuous data (heights, weights, incomes grouped into intervals), enter the midpoint of each class interval. For discrete data grouped into ranges (test scores 60-69, 70-79, etc.), use the midpoint of each range (64.5, 74.5, etc.). The math is identical. The only difference is interpretation: for continuous data, the midpoint is an approximation; for discrete grouped data with uniform spacing, the midpoint is exact for equally-spaced integer ranges.

What causes large standard deviation in grouped frequency data?

A large standard deviation indicates high spread of values around the mean. It can result from: (1) heavy tails with many values far from the centre class; (2) a bimodal distribution where two separated classes dominate; (3) a wide range of class midpoints with significant frequencies at both extremes; or (4) genuinely high variability in the underlying data. A small standard deviation indicates values cluster tightly around the mean.

📌 Quick Tips

💡The midpoint for each class interval is (lower boundary + upper boundary) divided by 2. For the class 20-30, the midpoint is 25. Always use midpoints, not boundaries, in this calculator.

💡Use sample standard deviation (s) when your frequency table represents a sample drawn from a larger population. Use population standard deviation (sigma) when the table covers the entire population.

💡The coefficient of variation (CV = s divided by x-bar times 100) lets you compare variability between datasets with different units or different means. A lower CV means less relative spread.

💡Increasing the number of class intervals reduces grouping error in the mean and SD. The standard deviation of grouped data is always an approximation because all values within a class are assumed equal to the midpoint.

💡If one class has a much higher frequency than others, that class dominates the mean. A very high-frequency class near the centre pulls the mean toward it and reduces the standard deviation.