Standard Deviation Calculator

This free standard deviation calculator finds the sample and population standard deviation of any data set, with the full step-by-step working, a chart, z-scores and the exact TI-84 keys for the same numbers. Results update as you type.

How to Use the Standard Deviation Calculator

  1. Enter your data. Type or paste numbers separated by commas, spaces, tabs or new lines. You can paste a column straight from Excel or Google Sheets.
  2. Pick the data type. Use List of numbers for raw data, Value + frequency for a frequency table (for example 3, 5 means the value 3 appears 5 times), or Grouped classes for intervals such as 10-20, 4.
  3. Choose sample or population. Both answers are always shown. Your choice decides which one is highlighted and which one the steps and z-scores use.
  4. Read the results. You get the standard deviation, variance, mean, median, mode, range, quartiles, IQR, standard error, coefficient of variation, sums and outliers. Use Decimals to change the rounding.
  5. Copy or share. Copy results copies a plain-text summary. Copy share link copies a link that reopens the calculator with your data.

Standard Deviation Formulas

Standard deviation measures how far values typically sit from the mean. A small standard deviation means the data are bunched together; a large one means they are spread out.

Population standard deviation: σ = √[ Σ(x − μ)² / N ]

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

Here x is each value, μ or x̄ is the mean, and N or n is the number of values. The variance is the same expression without the square root. A handy shortcut for the sum of squared deviations is SS = Σx² − (Σx)² / n; the calculator shows it as a check.

For a frequency table, each squared deviation is multiplied by its frequency f, and n = Σf. For grouped data, the class midpoint stands in for every value in the class, so the result is an estimate.

Sample vs Population Standard Deviation

Use the population formula when your data contain every member of the group you care about, such as the test scores of every student in one class when you only care about that class. Use the sample formula when your data are part of a bigger group and you want to estimate its spread, such as 50 surveyed customers out of thousands. Most homework and real research uses the sample version.

The sample standard deviation is always a little larger, because it divides by n − 1 instead of n. The gap shrinks as n grows. With only one value, the sample standard deviation is undefined, because n − 1 = 0.

Worked Example

Find the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

  1. Mean: (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5.
  2. Deviations from the mean: −3, −1, −1, −1, 0, 0, 2, 4. They always add up to 0, which is a good check.
  3. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16. Their sum is SS = 32.
  4. Population: σ² = 32 / 8 = 4, so σ = √4 = 2.
  5. Sample: s² = 32 / 7 ≈ 4.5714, so s = √4.5714 ≈ 2.1381.

This is the example loaded in the calculator above, so you can compare each step with its table.

How to Find Standard Deviation on a TI-84

The TI-84 Plus, TI-84 Plus CE and TI-83 Plus all use the same keys. You can follow along on our free TI-84 Plus CE calculator online.

  1. Press STAT, 1 (Edit…). If L1 has old data, arrow up onto the L1 heading and press CLEAR, ENTER.
  2. Type each value and press ENTER after it. Use the (−) key, not the minus key, for negative numbers.
  3. For a frequency table, put the values in L1 and the frequencies in L2.
  4. Press STAT, arrow right to CALC and press 1 (1-Var Stats). Set List to L1 (and FreqList to L2 if you used frequencies), choose Calculate and press ENTER.
  5. Read Sx for the sample standard deviation and σx for the population standard deviation. Scroll down for n, minX, Q1, Med, Q3 and maxX.

The calculator above builds these steps for your own data, including the numbers to type and what the 1-Var Stats screen should show. Its quartiles use the TI-84 method, so they match your calculator exactly.

Tips for Getting the Right Answer

  • Check which formula your teacher wants. Picking sample when the question means population (or the other way round) is the most common mistake.
  • Look at the skipped-entries note. A stray letter or symbol in pasted data is skipped and listed, so you know exactly what was counted.
  • Watch for outliers. One extreme value can inflate the standard deviation a lot. The calculator flags values outside the 1.5 × IQR fences and highlights z-scores beyond ±2.
  • Round at the end. Keep full precision until the final answer; use the Decimals control only for what you write down.

Frequently Asked Questions

Population standard deviation (σ) divides the sum of squared deviations by N, the number of values, and is used when your data include every member of the group. Sample standard deviation (s) divides by n − 1 and is used when your data are a sample from a bigger group. The calculator shows both side by side.

A sample’s values sit a little closer to their own mean than to the true population mean, so dividing by n would underestimate the spread. Dividing by n − 1 (Bessel’s correction) fixes that bias for the variance.

Yes. Copy a column or a row and paste it into the box. Tabs, new lines, commas, spaces and semicolons all work, and anything that is not a number is skipped and listed so you can check it.

Enter the data in L1 with STAT, 1. Then press STAT, arrow to CALC, choose 1 (1-Var Stats), set List to L1 and press ENTER. Sx is the sample standard deviation and σx is the population standard deviation.

Choose the Value + frequency tab and type one value and its frequency per line, such as 3, 5. For class intervals like 10–20, use the Grouped classes tab, which works from class midpoints.

This calculator uses the same quartile method as the TI-84: the median of the lower and upper halves, leaving out the middle value when n is odd. Excel’s QUARTILE.INC uses interpolation, so Q1 and Q3 can differ slightly. The standard deviation is the same either way.

Ti84CalcPro is not affiliated with or endorsed by Texas Instruments. TI-84 Plus CE is a trademark of Texas Instruments.