Statistics calculator

For ten test scores from 66 to 94, the statistics calculator gives a mean of 81.3, a median of 85, a mode of 85 and a sample standard deviation of 9.32.

Mean81.3

Sample standard deviation: 9.322
Working SS is the sum of squared deviations from the mean of 81.3.: s = sqrt(SS / (n - 1)) = sqrt(782.1 / 9) = sqrt(86.9) = 9.322
Sample variance: 86.9
Count: 10
Mean Sum of 813 over 10 values.: 81.3
Median The average of the two middle values.: 85
Mode: 85
Range: 28 (66 to 94)
If this were the whole population Dividing by n instead of n-1 gives 8.8436, which is 5.1% smaller. Use it only when these numbers are every case you care about, not a sample from something larger.: 8.8436
Quartiles Interquartile range 16: the middle half of the data. Unlike the standard deviation, it is barely affected by an extreme value.: Q1 72, median 85, Q3 88
Outliers By the 1.5 x IQR rule: anything below 48 or above 112.: none
Z-score of 94 1.36 standard deviations above the mean. If the data were normally distributed, about 91.3% of values would fall below it.: 1.362
Standard error of the mean The standard deviation divided by the square root of n: how much the sample mean itself would wobble between samples. Quartering it takes sixteen times the data.: 2.9479
Coefficient of variation Standard deviation as a share of the mean. The only one of these figures that compares the spread of two things measured in different units.: 11.47%
If the data were normal 95% within 62.656 to 99.944, and 99.7% within 53.3339 to 109.2661. That is the empirical rule, and it only holds for roughly bell-shaped data.: 68% within 71.978 to 90.622

Deviation from the mean

Value	Deviation	Squared
66	-15.3	234.09
70	-11.3	127.69
72	-9.3	86.49
78	-3.3	10.89
85	3.7	13.69
85	3.7	13.69
85	3.7	13.69
88	6.7	44.89
90	8.7	75.69
94	12.7	161.29
Total	0	782.1

The deviations always sum to zero; that is what the mean is. Squaring them before averaging is what stops them canceling, and it is also why the standard deviation is in the same units as the data while the variance is in units squared.
Non-numeric entries in the list are ignored rather than read as zero, so a header row pasted in with a column will not drag the mean down.

How the Standard Deviation Calculator works

Descriptive statistics summarize a list with two kinds of number: where its center is, and how spread out it is. The center is the mean, the median and the mode. The spread is the range, the quartiles, the variance and the standard deviation.

Mean = sum ÷ count. The ten scores add up to 813, so the mean is 813 ÷ 10 = 81.3. The median is the middle value once the list is sorted: 66, 70, 72, 78, 85, 85, 85, 88, 90, 94. With an even count it is the average of the two middle values, (85 + 85) ÷ 2 = 85. The mode is the most frequent value, 85, which appears three times. The range is the largest minus the smallest, 94 - 66 = 28.

The sample standard deviation is s = √(Σ(x - mean)² ÷ (n - 1)). The squared distances from 81.3 add up to 782.1, and 782.1 ÷ 9 = 86.9 is the sample variance. Its square root, 9.32, is the standard deviation, in the same units as the scores.

Quartiles split the sorted list into quarters. Q1 is 72 and Q3 is 88, so the interquartile range is 16. The 1.5 × IQR rule flags anything below 48 or above 112 as an outlier, and none of these scores is one.

When the mean sits below the median, as 81.3 does against 85 here, the low scores are pulling it down and the data is skewed left. The median is the better center for skewed data; the mean is the one the variance and standard deviation are built on.

Standard Deviation Calculator FAQ

Should I use the sample or the population standard deviation?

Sample, when the numbers are drawn from a larger group you want to describe. Population, when they are the whole group. Treated as a whole class, these ten scores have a population standard deviation of 8.84 instead of 9.32.

What is a z-score?

How many standard deviations a value sits from the mean: (value - mean) ÷ SD. A score of 94 is (94 - 81.3) ÷ 9.322 = 1.36 standard deviations above the mean.

Can a list have more than one mode?

Yes. If two values tie for most frequent, both are modes and the data is bimodal. If no value repeats, there is no mode.

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