Variance calculation is off I'm trying to find the variance of a set of numbers $90,90,80,100,99,81,98,82$. When I do it manually and when I use online calculator sites I get the answer $61.25$ but when i put the data into a stat list and use the variance( button on my TI-84 I get the answer $70$. My calculator works and I'm absolutely positive I entered the list right. What's wrong?
 A: There are statsticians who prefer using $n-1$ rather than $n$ when computing variance. Your sum of squared differences fromm the mean is indeed 490. Divide by either 8 or 7
https://en.wikipedia.org/wiki/Bessel%27s_correction
A: Many statistics and probability books make a distinction between two kinds of 'variances'.
(a) Variance of a finite population $\sigma^2 = \frac 1 N \sum_{i=1}^N (X_i - \mu)^2,$ where $\mu = \frac 1 N \sum_{i=1}^N X_i$ is the population mean and $N$ the population size. For your $N=8$ values, $\sigma^2 = 61.25.$
(b) Variance of a sample from an infinite population $S^2 = \frac{1}{n-1} 
\sum_{i=1}^n (X_i = \bar X)^2,$ where $\bar X = \frac 1 n
\sum_{i=1}^n X_i$ is the the sample mean and $n$ is the sample size.
For your $n = 8$ values, $S^2 = 70.$
A sample variance defined with $n-1$ in the denominator has the advantage
that $E(S^2) = \sigma^2.$ where the population variance $\sigma^2$ exists.
Accordingly, one says that $S^2$ is an 'unbiased' estimator of $\sigma^2.$
[For further discussion and a formal proof of unbiasedness, see this page (especially answers by me and @Vivek)--among others on this site and elsewhere online.]

Some calculators give a choice whether to compute a standard deviation (square root of variance) for entered values regarded as a population or as a sample.
Sometimes buttons are labeled $\sigma_n$ and $\sigma_{n-1}.$ Neither is perfect
notation, but the choice is clear. (On the TI-84, the distinction is made
between the sample standard deviation 
'$\text{Sx}$' and the population standard deviation '$\sigma\text{x}$'; see p206 of the Guide Book.)
In R statistical software, the function var implements the formula for the sample variance (denominator $n-1).$ 
x = c(90,90,80,100,99,81,98,82);  var(x)
## 70        # variance of vector x treated as a sample
N = length(x);  (N-1)*var(x)/N
## 61.25     # adjustment to treat vector x as a population


Note: A brief simulation in R illustrates the unbiasedness of $S^2$
as an estimator of $\sigma^2.$ A million samples of size $n = 1000$ are sampled
from a normal population with mean $\mu = 50$ and standard deviation $\sigma = 5,$ variance $\sigma^2 = 25.$ The sample variance $S^2$ (denoted v in the program) is found for each sample. The average of the one million sample variances
approximates $E(S^2)$ accurate to several significant digits. (Even for samples of size $n = 1000,$
there is considerable variability among the values of $S^2;$ in this particular
simulation the smallest sample variance was 19.82 and the largest was 30.87.
"Variances are very variable.")
set.seed(630);  m = 10^6
v = replicate(m,  var(rnorm(1000, 50, 5)))  # vector of 10^6 sample variances
mean(v)
##  25.00077  # aprx E(v) = 25.

