Standard Deviation Calculator: Mean, Median, Mode and Steps
Mean
75.9
Median
77.5
Mode
85
appears 2 times
Range
32
90 − 58
Q1 / Q3
69 / 85
IQR
16
no outliers (1.5 × IQR rule)
Sorted: 58, 64, 69, 72, 77, 78, 81, 85, 85, 90
Sample standard deviation (s)
10.1154
- Mean
- 75.9
- Median
- 77.5
- Mode
- 85 (×2)
- Sample SD (s)
- 10.1154
- Population SD (σ)
- 9.5964
- Sample variance (s²)
- 102.3222
- Population variance (σ²)
- 92.09
- Sum
- 759
- Min / Max
- 58 / 90
Step by step: sample standard deviation
The working you'd write in an exam, with every value.
- 1. Find the mean. x̄ = Σx ÷ n = 759 ÷ 10 = 75.9
- 2. Subtract the mean from each value and square the result.
| x | x − x̄ | (x − x̄)² |
|---|---|---|
| 72 | -3.9 | 15.21 |
| 85 | 9.1 | 82.81 |
| 64 | -11.9 | 141.61 |
| 90 | 14.1 | 198.81 |
| 78 | 2.1 | 4.41 |
| 85 | 9.1 | 82.81 |
| 69 | -6.9 | 47.61 |
| 58 | -17.9 | 320.41 |
| 81 | 5.1 | 26.01 |
| 77 | 1.1 | 1.21 |
| Σ = 759 | 0 | 920.9 |
- 3. Divide by n − 1 to get the variance. s² = Σ(x − x̄)² ÷ (10 − 1) = 920.9 ÷ 9 = 102.3222
- 4. Take the square root. s = √102.3222 = 10.1154
- Shortcut formula checkΣx² ÷ n − x̄², the same variance by the shortcut form (divides by n)92.09
- Population variance σ²92.09
On a Casio fx-97SG X, enter the data in STAT mode (1-Var): σx is the population SD and sx the sample SD. The O-Level formula list's standard deviation divides by n (Σf), the population form. Use s when the data is a sample used to estimate a population, as in A-Level H2 statistics.
How this works
The mean is the sum divided by the count, the median is the middle value once sorted (the average of the two middle values when the count is even), and the mode is the value that appears most often; a list where every value appears once has no mode, and ties show every mode. The range is the largest value minus the smallest.
Standard deviation measures spread around the mean. Square each value's distance from the mean, add them up, divide, and take the square root. Divide by n for the population standard deviation (σ, the form in the O-Level formula list, and σx on a Casio). Divide by n − 1 for the sample standard deviation (s, sx on a Casio, STDEV.S in Excel), used when the data is a sample from a larger group. Variance is the standard deviation squared.
Quartiles split the sorted data into four. The default “median of halves” method takes the median of the lower and upper halves, leaving out the middle value when the count is odd, as on a TI-84 and in most school textbooks. The Excel QUARTILE.INC option interpolates at position (n − 1) × p instead, so the two methods can differ slightly on small data sets. The interquartile range is Q3 − Q1, and values more than 1.5 × IQR outside the quartiles are flagged as outliers.
FAQs
How do I calculate standard deviation by hand?
Find the mean, subtract it from each value, square the results, add them up, divide by n (population) or n − 1 (sample), then take the square root. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the squared deviations add to 32, and the population SD is √(32 ÷ 8) = 2.
What is the difference between sample and population standard deviation?
Population SD divides by n and describes the data set itself; sample SD divides by n − 1 and estimates the spread of a larger population from a sample. For the scores 72, 85, 64, 90, 78, 85, 69, 58, 81 and 77, the population SD is 9.5964 and the sample SD is 10.1154.
What is the mean, median and mode of 72, 85, 64, 90, 78, 85, 69, 58, 81, 77?
The mean is 75.9 (759 ÷ 10), the median is 77.5 (the average of 77 and 78), and the mode is 85, which appears twice. The range is 90 − 58 = 32.
What if there are two modes?
The data is bimodal and both values are modes. The calculator lists every value that shares the highest count; if every value appears once, there is no mode.