You should understand the difference between the parameters and properties of a distribution and the estimators for these parameters and properties. For instance

• The true mean, $\mu = E[X]$ is the the expected value of a stochastic variable $X$ and can not be calculated exactly.
• The sample mean, $m = \sum{x_i/n}$ with $x_i$ your observations of $X$ is the usual estimator for $\mu$.

Chapters in text books and whole scientific articles discuss the quality of estimators. For variance

• The true variance is $\sigma^2 = E[(X-\mu)^2]$
• The sample variance actually is $\frac{1}{n} \sum{(x_i - m)^2}$, but it has a tendency to be smaller than $\sigma^2$, therefore it is said to be biassed. This is related to the fact that $m$ itself is estimated from the same sample.
• The usual estimator, $s^2 = \frac{1}{n-1} \sum{(x_i - m)^2}$, does not have this disadvantage.

For skewness

• The true skewness of the stochastic variable is $\gamma_1 = E[(\frac{X-\mu}{\sigma})^3]$

• The sample skewness is $\frac{1}{n} \sum(\frac{x_i-m}{s})^3$, but again, it is biassed.

• The usual estimator is $\frac{n}{(n-1)(n-2)} \sum(\frac{x_i-m}{s})^3$, in which $s$ is of course the square root of the estimator for the variance.

Have fun implementing this. For further discussion, you might consult

• Wikipedia's article on Bias of an estimator
• this discussion of SAS on descriptive statistics or
• this Cross Validate question on skewness and kurtosis.

You should understand the difference between the parameters and properties of a distribution and the estimators for these parameters and properties. For instance

• The true mean, $\mu = E[X]$ is the the expected value of a stochastic variable $X$ and can not be calculated exactly.
• The sample mean, $m = \sum{x_i/n}$ with $x_i$ your observations of $X$ is the usual estimator for $\mu$.

Chapters in text books and whole scientific articles discuss the quality of estimators. For variance

• The true variance is $\sigma^2 = E[(X-\mu)^2]$
• The sample variance actually is $\frac{1}{n} \sum{(x_i - m)^2}$, but it has a tendency to be smaller than $\sigma^2$, therefore it is said to be biassed. This is related to the fact that $m$ itself is estimated from the same sample.
• The usual estimator, $s^2 = \frac{1}{n-1} \sum{(x_i - m)^2}$, does not have this disadvantage.

For skewness

• The true skewness of the stochastic variable is $\gamma_1 = E[(\frac{X-\mu}{\sigma})^3]$

• The sample skewness is $\frac{1}{n} \sum(\frac{x_i-m}{s})^3$, but again, it is biassed.

• The usual estimator is $\frac{n}{(n-1)(n-2)} \sum(\frac{x_i-m}{s})^3$, in which $s$ is of course the square root of the estimator for the variance.

Have fun implementing this. For further discussion, you might consult

• Wikipedia's article on Bias of an estimator
• this discussion of SAS on descriptive statistics or
• this Cross Validate question on skewness and kurtosis.

You should understand the difference between the parameters and properties of a distribution and the estimators for these parameters and properties. For instance

• The true mean, $\mu = E[X]$ is the the expected value of a stochastic variable $X$ and can not be calculated exactly.
• The sample mean, $m = \sum{x_i/n}$ with $x_i$ your observations of $X$ is the usual estimator for $\mu$.

Chapters in text books and whole scientific articles discuss the quality of estimators. For variance

• The true variance is $\sigma^2 = E[(X-\mu)^2]$
• The sample variance actually is $\frac{1}{n} \sum{(x_i - m)^2}$, but it has a tendency to be smaller than $\sigma^2$, therefore it is said to be biassed. This is related to the fact that $m$ itself is estimated from the same sample.
• The usual estimator, $s^2 = \frac{1}{n-1} \sum{(x_i - m)^2}$, does not have this disadvantage.

For skewness

• The true skewness of the stochastic variable is $\gamma_1 = E[(\frac{X-\mu}{\sigma})^3]$

• The sample skewness is $\frac{1}{n} \sum(\frac{x_i-m}{s})^3$, but again, it is biassed.

• The usual estimator is $\frac{n}{(n-1)(n-2)} \sum(\frac{x_i-m}{s})^3$, in which $s$ is of course the square root of the estimator for the variance.

Have fun implementing this. For further discussion, you might consult

• Wikipedia's article on Bias of an estimator
• this discussion of SAS on descriptive statistics or
• this Cross Validate question on skewness and kurtosis.

You should understand the difference between the parameters and properties of a distribution and the estimators for these parameters and properties. For instance

• The true mean, $\mu = E[X]$ is the the expected value of a stochastic variable $X$ and can not be calculated exactly.
• The sample mean, $m = \sum{x_i/n}$ with $x_i$ your observations of $X$ is the usual estimator for $\mu$.

Chapters in text books and whole scientific articles discuss the quality of estimators. For variance

• The true variance is $\sigma^2 = E[(X-\mu)^2]$
• The sample variance actually is $\frac{1}{n} \sum{(x_i - m)^2}$, but it has a tendency to be smaller than $\sigma^2$, therefore it is said to be biassed. This is related to the fact that $m$ itself is estimated from the same sample.
• The usual estimator, $s^2 = \frac{1}{n-1} \sum{(x_i - m)^2}$, does not have this disadvantage.

For skewness

• The true skewness of the stochastic variable is $\gamma_1 = E[(\frac{X-\mu}{\sigma})^3]$

• The sample skewness is $\frac{1}{n} \sum(\frac{x_i-m}{s})^3$, but again, it is biassed.

• The usual estimator is $\frac{n}{(n-1)(n-2)} \sum(\frac{x_i-m}{s})^3$, in which $s$ is of course the square root of the estimator for the variance.

Have fun implementing this. For further discussion, you might consult

• Wikipedia's article on Bias of an estimator
• this discussion of SAS on descriptive statistics or
• this Cross Validate question on skewness and kurtosis.