How to Calculate Standard Deviation Step by Step

Short answer: to calculate standard deviation, find the mean, subtract it from each value and square the result, average those squared differences, then take the square root. For the test scores 66, 74, 82, 88, and 90, the mean is 80, the squared differences add up to 400, and the population standard deviation works out to about 8.9. That’s the whole method, but the details trip people up: whether to divide by N or n - 1, what the answer actually means, and which spreadsheet function matches which formula. This guide on how to calculate standard deviation walks through every step by hand, shows both formulas on the same data set, and decodes the Excel and calculator buttons.

Want to skip the arithmetic? The standard deviation tool accepts any data set, shows both the sample and population results, and lists the intermediate steps so you can follow the work.

What Is Standard Deviation?

Standard deviation measures how spread out a data set is around its mean. A low standard deviation means the values huddle close to the average; a high one means they’re scattered far from it. It’s the single most used number in statistics after the mean itself, because an average alone hides almost everything interesting about data.

Two basketball players make the point. Both average exactly 20 points a game over five games, yet they’re nothing alike:

Player	Game scores	Mean	Sample standard deviation
Player A	18, 20, 22, 19, 21	20	1.6
Player B	5, 35, 10, 30, 20	20	12.7

Player A delivers 20 points like clockwork. Player B might drop 35 or might barely score. The mean can’t tell those stories apart; the standard deviation can. That’s why it shows up everywhere spread matters: grading curves, quality control, weather records, investment risk, lab results. In finance it literally is the standard measure of risk. Two funds can both average 8% a year, but the one whose annual returns swing with a standard deviation of 18% will hand you a very different ride than the one sitting at 5%.

The units always match the original data. If you’re measuring dollars, the standard deviation is in dollars. If it’s minutes, the answer is in minutes. That’s one of its biggest advantages over variance, which we’ll get to later.

What Is the Standard Deviation Formula?

The standard deviation formula comes in two versions, and the only difference is one number in the denominator.

Population standard deviation: σ = √( Σ(x - μ)² / N )

Sample standard deviation: s = √( Σ(x - x̄)² / (n - 1) )

In words: subtract the mean from each value, square each result, add the squares up, divide by N (population) or n - 1 (sample), and take the square root. The symbols scare people more than the math does, so here’s the full decoder:

Symbol	Name	What it means
σ	sigma (lowercase)	Population standard deviation
s	s	Sample standard deviation
σ², s²	sigma squared, s squared	Population and sample variance
μ	mu	Population mean
x̄	x-bar	Sample mean
Σ	sigma (uppercase)	“Sum of” - add up everything that follows
N	big N	Number of values in a whole population
n	little n	Number of values in a sample

So Σ(x - μ)² just reads as “take each value, subtract the mean, square it, and add all of those up.” Every calculation in this guide is that phrase turned into arithmetic.

How to Calculate Standard Deviation by Hand?

Here’s how to find standard deviation in six steps, worked on a real data set: five quiz scores of 66, 74, 82, 88, and 90.

Step 1: find the mean. Add the values and divide by how many there are. 66 + 74 + 82 + 88 + 90 = 400, and 400 / 5 = 80. (For messier data sets, the mean, median and mode tool does this part instantly.)

Step 2: subtract the mean from each value. These are the deviations. They must add up to zero, which is a free arithmetic check.

Step 3: square each deviation. Squaring makes every number positive and weights far-away values more heavily.

Score	Deviation (score - 80)	Squared deviation
66	-14	196
74	-6	36
82	2	4
88	8	64
90	10	100
Total	0 ✓	400

Step 4: add the squared deviations. 196 + 36 + 4 + 64 + 100 = 400.

Step 5: divide to get the variance. Divide by N = 5 if these five scores are the entire group you care about: 400 / 5 = 80. Divide by n - 1 = 4 if they’re a sample from a bigger group: 400 / 4 = 100.

Step 6: take the square root. √80 ≈ 8.94 for the population version, √100 = 10 for the sample version.

Done. If this was the whole class, the scores typically sit about 9 points from the average of 80. If it was five students sampled from a bigger class, the best estimate of spread is 10 points. You can check every step against your own numbers and see the same table generated automatically.

Should You Use the Population or Sample Formula?

This is the step that splits answers, so it deserves its own decision. Use the population formula when your data set is every value you care about: all 30 students in your class, all 12 months of your own spending, every unit a machine produced today. Use the sample formula when your data stands in for a bigger group you couldn’t measure: 50 shoppers surveyed out of thousands, 20 soil readings from a whole field.

Why n - 1? A sample’s values are always a little closer to their own average than to the true population mean, so dividing by n underestimates the spread. Dividing by n - 1, known as Bessel’s correction, stretches the answer just enough to compensate. Here’s the same data set from above run through both formulas side by side:

Step	Population (σ)	Sample (s)
Sum of squared deviations	400	400
Divide by	N = 5	n - 1 = 4
Variance	80	100
Standard deviation	8.94	10.00

Same data, answers about 12% apart. The gap shrinks as the data set grows: with 100 values, dividing by 99 instead of 100 barely moves the result. With 5 values, it clearly matters.

When in doubt, use the sample formula. Most real-world data is a sample of something larger, which is why spreadsheets default to it. Statistics classes usually tell you which one they want, and grading keys do punish the wrong denominator.

What Does Standard Deviation Actually Tell You?

A standard deviation on its own is just a number. Its power shows up when your data is roughly bell-shaped, because then the empirical rule, also called the 68-95-99.7 rule, turns it into predictions:

Range	Share of values	Quiz example (mean 80, s = 10)
Within 1 standard deviation	about 68%	70 to 90
Within 2 standard deviations	about 95%	60 to 100
Within 3 standard deviations	about 99.7%	50 to 110

So if a large class averages 80 with a standard deviation of 10, roughly two thirds of students scored between 70 and 90, and a score below 60 puts a student in about the bottom 2.5%. Counting how many standard deviations a value sits from the mean is exactly what a z-score measures, and it’s how curved grades and growth charts are built.

The other interpretation trick is judging whether a standard deviation is “big.” Raw size means nothing without the mean next to it, so divide the standard deviation by the mean to get the coefficient of variation (CV), a percentage you can compare across any two data sets:

Data set	Mean	Standard deviation	CV	Reading
Cafe A wait time	4.0 min	0.5 min	12.5%	Consistent, predictable
Cafe B wait time	4.0 min	3.0 min	75%	Wildly variable
Salaried paycheck	$1,200/wk	$50	4.2%	Very stable income
Gig-work income	$1,200/wk	$400	33.3%	Feast or famine

Both cafes advertise the same average wait, but only one of them ever takes 10 minutes. The CV also lets you compare spread across different units entirely, like heights in inches against weights in pounds, where raw standard deviations can’t be compared at all. Since CV is just a percent calculation, the same method from our guide on how to calculate percentage applies: part over whole, times 100.

How Do You Find Standard Deviation in Excel or on a Calculator?

Nobody computes standard deviation by hand twice. The catch is that every tool offers both formulas, and the buttons don’t announce which is which. Here’s the decoder:

Tool	Sample (÷ n - 1)	Population (÷ N)
Excel	=STDEV.S(range) or older =STDEV	=STDEV.P(range) or older =STDEVP
Google Sheets	=STDEV(A1:A20)	=STDEVP(A1:A20)
TI-83/TI-84	Sx in 1-Var Stats	σx in 1-Var Stats
Most phone/stat apps	Usually labeled s or “sample”	Usually labeled σ or “population”

In Excel or Sheets, type your values into a column, click an empty cell, and enter =STDEV.S(A1:A20) with your own range. On a TI-84, press STAT, enter the data in L1, then STAT again, over to CALC, choose 1-Var Stats, and read both Sx and σx from the output screen. The calculator prints both every time; picking the right one is on you.

Two habits prevent most spreadsheet errors. First, spot-check the range: a stray blank or text cell silently changes the count. Second, sanity-check the output against the rule that nearly all values should fall within 3 standard deviations of the mean. If =STDEV.S returns 45 on data that only spans 20 units, the range grabbed something it shouldn’t have.

How Does an Outlier Change the Standard Deviation?

Because every deviation gets squared, values far from the mean don’t just add to the standard deviation, they dominate it. One wild data point can swamp everything else, and seeing that happen once will change how you read summary statistics forever.

Take the quiz scores from earlier and change a single number: the 90 becomes a 190 (say, a typo while entering data). Watch what happens:

	Original data (66, 74, 82, 88, 90)	With outlier (66, 74, 82, 88, 190)
Mean	80	100
Sum of squared deviations	400	10,400
Sample variance	100	2,600
Sample standard deviation	10	51

One typo moved the mean by 20 points and multiplied the standard deviation by five. The squaring step is the culprit: the outlier’s deviation of 90 becomes 8,100 when squared, which is more than twenty times the combined contribution of the other four values.

Two practical lessons follow. First, always scan your data before trusting the output; a standard deviation that looks absurdly large is often a data-entry error, not real spread. Second, for data with legitimate extreme values, like household incomes or home prices, the standard deviation can paint a misleading picture of the typical case. Statisticians reach for the interquartile range in those situations, since it ignores the extremes entirely. The standard deviation shines on roughly symmetric data and struggles on heavily skewed data, and knowing that boundary is half of using it well.

Standard Deviation vs Variance: What’s the Difference?

Variance is the step you pass through on the way to standard deviation: it’s the average of the squared deviations before you take the square root. The two numbers carry the same information but wear different units, and that changes what each is good for:

	Variance	Standard deviation
Symbol	σ² or s²	σ or s
Formula relation	Average of squared deviations	Square root of variance
Units	Original units squared	Original units
Quiz example	100 points²	10 points
Best for	Statistical theory, ANOVA, combining spreads	Interpreting and reporting real data

“Points squared” means nothing to a reader, which is why reports quote standard deviation. Variance earns its keep inside statistics, where its math behaves better: variances of independent variables add cleanly, standard deviations don’t. For everyday use the rule is simple: compute through variance, report the standard deviation.

What Are the Most Common Standard Deviation Mistakes?

The method is mechanical, but four specific slips produce most wrong answers. Each one has a quick fix:

Mistake	What it looks like	Fix
Wrong denominator	Dividing by n for a sample (or n - 1 for a population)	Whole group in hand: N. Subset standing in for more: n - 1
Reporting variance	Stopping at 100 instead of taking √100 = 10	The last step is always a square root; check the units make sense
Dropping negatives	Treating -14 as 14 before squaring, or losing the sign in Step 2	Deviations must sum to 0 before squaring; if they don’t, recheck
Comparing raw SDs across scales	”SD of 50 beats SD of 5” when one mean is 1,000 and the other is 10	Divide by the mean first and compare CVs instead

The deviation-sum check in row three is worth making a habit. It costs five seconds, and it catches subtraction slips before they poison every later step. The units check in row two is just as fast: a standard deviation should always be a plausible distance in the data’s own units, so a “standard deviation” of 400 on quiz scores that range from 66 to 90 means a square root got skipped.

Standard deviation rewards doing it by hand exactly once, so the formula stops being a mystery, and then letting software carry it from there. Work through your own numbers with the step-by-step solver, compare the sample and population answers, and you’ll never wonder which button your calculator wants you to press again.