6.11 Random Variables, Distributions & Descriptive Statistics

Measures of Central Tendency

Three averages are named in the syllabus, and each answers a different question.

Arithmetic mean. For raw data $\bar{x} = \dfrac{\sum x_i}{n}$; for grouped data $\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}$. The mean uses every observation, which also makes it sensitive to outliers.

Median. The middle value once the data are ordered. For $n$ odd it is the $\left(\dfrac{n+1}{2}\right)$th item; for $n$ even it is the average of the two central items. The median is unaffected by extreme values, which is why it is preferred for skewed data such as repair times.

Mode. The most frequently occurring value. A data set may have no mode, one mode, or several.

For a moderately skewed distribution the three are linked by the empirical relation

$\text{Mode} \approx 3\,\text{Median} - 2\,\text{Mean}$

In a symmetric distribution such as the normal, all three coincide.

Measures of Dispersion

A computationally convenient identity is

$\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2 = E(X^2) - \left[E(X)\right]^2$

For a sample rather than a population, the denominator becomes $n - 1$ (Bessel's correction), which produces an unbiased estimate of the population variance.

Worked example. For the data 2, 4, 4, 4, 5, 5, 7, 9: the mean is $40/8 = 5$; the sum of squares is $4+16+16+16+25+25+49+81 = 232$, so $\sigma^2 = 232/8 - 25 = 29 - 25 = 4$ and $\sigma = 2$. The median is $(4+5)/2 = 4.5$ and the mode is 4.

Random Variables

A random variable assigns a number to each outcome of an experiment. It is discrete if it takes countable values (number of defective bearings in a batch) and continuous if it takes any value in an interval (bore diameter of a cylinder).

For a discrete variable the expectation and variance are

$E(X) = \sum p_i x_i, \qquad V(X) = E(X^2) - [E(X)]^2$

Useful operational rules: $E(aX + b) = aE(X) + b$, while $V(aX + b) = a^2 V(X)$. Adding a constant shifts the mean but leaves the spread untouched — a point objective items test directly.

The Three Named Distributions

Binomial distribution

Applies to $n$ independent trials, each with the same success probability $p$ (and $q = 1 - p$):

$P(X = r) = \binom{n}{r}p^{r}q^{\,n-r}, \qquad r = 0, 1, \ldots, n$

$\text{Mean} = np, \qquad \text{Variance} = npq$

Because $q < 1$, the variance of a binomial variable is always less than its mean — a quick consistency check on any answer.

Example. A drill bit has a 10% chance of failing per shift. In 5 shifts, the probability of exactly one failure is $\binom{5}{1}(0.1)(0.9)^4 = 5(0.1)(0.6561) = 0.328$.

Poisson distribution

The limiting case of the binomial when $n \to \infty$, $p \to 0$ with $\lambda = np$ finite. It models rare events in a fixed interval — conveyor belt failures per month, flaws per metre of weld:

$P(X = r) = \frac{e^{-\lambda}\lambda^{r}}{r!}$

$\text{Mean} = \text{Variance} = \lambda$

The equality of mean and variance is the distribution's signature; a data set whose sample mean and variance are close is well modelled as Poisson.

Example. If a longwall face averages 2 unplanned stoppages per week, the probability of no stoppage in a given week is $e^{-2} = 0.135$.

Normal distribution

The continuous bell curve with density

$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp\!\left[-\frac{(x-\mu)^2}{2\sigma^2}\right]$

It is symmetric about $\mu$, where mean, median and mode all coincide, and the total area under the curve is 1. The standard normal variate is

$z = \frac{x - \mu}{\sigma}$

which has mean 0 and standard deviation 1. The empirical rule gives the areas worth memorising:

The last row is the basis of the $\pm 3\sigma$ control limits used in statistical process control and of the six-sigma quality vocabulary.

Comparison Table

Sampling

The syllabus also names sampling theorems. The central result is that the sample mean of a large sample is approximately normally distributed about the population mean with standard error

$\text{SE} = \frac{\sigma}{\sqrt{n}}$

regardless of the shape of the parent population. Quadrupling the sample size therefore halves the standard error — the reason inspection plans show diminishing returns as sample size grows.