Statistics
Paste values or use a frequency table to see how far your data typically sits from its mean.
Calculator
Your 8 values sit on average 2 from the mean of 5 using the population divisor. In your data 6 of 8 (75%) fall within one standard deviation; the 68-95-99.7 rule only belongs to roughly normal data.
Read 8 numbers.
import math
data = [2, 4, 4, 4, 5, 5, 7, 9]
data.sort()
mean = sum(data) / len(data)
mid = len(data) // 2
median = data[mid] if len(data) % 2 else (data[mid - 1] + data[mid]) / 2
print(f"count={len(data)}")
print(f"mean={mean}")
print(f"median={median}")
variance = sum((x - mean) ** 2 for x in data) / (len(data))
print(f"variance={variance}")
print(f"standard_deviation={math.sqrt(variance)}")
Both snippets print count, mean, median, variance and population standard deviation for the current values.
Print Room
Name: __________________ Date: ____________ Data: 2, 4, 4, 4, 5, 5, 7, 9
| Value | x - mean | (x - mean)^2 |
|---|---|---|
| 2 | −3 | 9 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
Totals: sum 40, squared deviations 32, population SD 2.
Reference: population variance = sum squared deviations ÷ n. Sample variance = sum squared deviations ÷ (n-1). Standard deviation = square root of variance.
Education
Standard deviation starts with the mean, then asks how far each value sits from that centre. For your current data the mean is 5. The value 2 has a deviation of −3 and a squared deviation of 9. Squaring is why distances below the mean do not cancel distances above it; a value 10 units away contributes four times as much variance as a value 5 units away.
Population SD treats the list as the whole population and divides by n. Sample SD treats the list as evidence about a larger population and divides by n-1. On the classic data set 2, 4, 4, 4, 5, 5, 7, 9, population variance is 4 and population SD is 2. The same squared deviations divided by 7 instead of 8 give sample SD about 2.13809. That larger answer is Bessel's correction: it compensates for the fact that a sample mean is pulled toward the sample values.
Variance and standard deviation carry the same information in different units. Variance is useful inside formulas, but it is in squared units. If delivery times are measured in minutes, variance is square minutes; SD returns to minutes. Standard error answers a different question again: it estimates uncertainty in the mean, not spread among individual deliveries. This calculator shows both so a report can say whether values vary a lot, whether the estimated mean is precise, or both.
The 68-95-99.7 rule belongs to roughly normal data. The overlay on the histogram is scaled to your mean, standard deviation and bin width, but the real check is counted from your actual data: 6 of 8 values fall within 1 SD, 8 within 2 SD and 8 within 3 SD. Outliers matter because squared deviations grow fast. In [1, 2, 3, 4, 100], the sample SD is 43.6177 even though four values are packed near 1 to 4.
In a classroom test-score example, a mean of 80 with an SD of 4 means most scores are fairly close to 80; a mean of 80 with an SD of 18 means the same average hides a much wider class. In delivery-time data, a low SD means customers can plan around a dependable arrival window, while a high SD means the mean alone is not enough. In manufacturing, SD is often watched because it tells whether the process is clustering around its target or drifting into wide variation. The unit stays the same as the original measure, which is why SD is easier to explain to a non-specialist than variance.
Frequency tables are useful when repeated values would make the raw list long. A table with value 4 and count 3 is the same as typing 4, 4, 4, and this calculator expands it before computing the same shared summary. That also gives you a quick audit path: the count shown under the frequency input should equal the sum of the count column. Zero can be a valid data value, but a zero count contributes no observations. Negative values are also valid for data such as temperatures or profit/loss, because standard deviation only needs numeric distances from the mean.
Use the z-score lookup when you need to compare one observation against the rest of its own group. A z score of 0 is exactly at the mean, +1 is one standard deviation above it, and -2 is two standard deviations below it. Z scores are dimensionless, so a test score, a delivery time and a lab measurement can be described with the same language, but only after each is compared with its own mean and SD.
Common mistakes: mixing sample and population SD without naming the divisor; calling standard error the spread of individuals; using a normal-curve claim on skewed data; deleting outliers without a documented reason; and rounding halfway through the working table. Keep full precision while calculating and round only for display.
Use population SD when your list is the whole group you care about. Use sample SD when the list is a sample and you want to estimate the spread of a larger group; the divisor is n-1.
Every deviation is squared before averaging, so the variance cannot be negative. Standard deviation is the square root of that variance.
It is a typical distance from the mean. If the data is roughly bell-shaped, about 68% of values often fall within one SD, but the calculator also counts your actual data.
Yes. Squaring deviations gives far-away values a large effect, so a single outlier can lift variance and SD sharply.
Standard error is sample SD divided by the square root of n. It describes uncertainty in the sample mean, not spread among individual observations.
Sample SD divides by n-1. With one value, n-1 equals zero, so the calculator states that sample SD needs at least two values.
Questions people ask
Use population SD when your list is the whole group you care about. Use sample SD when the list is a sample and you want to estimate the spread of a larger group; the divisor is n-1.
Every deviation is squared before averaging, so the variance cannot be negative. Standard deviation is the square root of that variance.
It is a typical distance from the mean. If the data is roughly bell-shaped, about 68% of values often fall within one SD, but the calculator also counts your actual data.
Yes. Squaring deviations gives far-away values a large effect, so a single outlier can lift variance and SD sharply.
Standard error is sample SD divided by the square root of n. It describes uncertainty in the sample mean, not spread among individual observations.
Sample SD divides by n-1. With one value, n-1 equals zero, so the calculator states that sample SD needs at least two values.
Certification notes
The formulas follow standard descriptive-statistics definitions for mean, variance, sample correction, standard deviation and standard error. Reviewed 2026-09-22.
Standard deviation describes numeric spread, not causation or data quality. The normal-curve overlay is a visual comparison only; the calculator counts your actual values inside each SD band so skewed or clumpy data is not forced to behave like a bell curve.
Use the canonical URL, the page title “Standard Deviation Calculator - CalculationTime”, and the review date 2026-09-22.