You've almost certainly heard about sampling, about the mean, the median,
and the standard deviation in your prior courses and work .
We will quickly explain these terms in this video, and add a few a other new terms.
The terminology we consider here is standard in the statistical community.
Important terminology items are highlighted in a purple colour in the slides and video.
Firstly, the idea of a population is that there is large collection
of potential measurements we could make.
That large number is capital N. This population represents all the data possible.
For example, the thickness of the wooden boards produced on our process last year:
we could go measure every single board from that time period.
If we did so, that would be our population.
In most instances, measuring the entire population is expensive, and impractical.
Next we introduce the idea of a sample.
If we go to our population and measure a subset, a much smaller number, lowercase n,
we can get a sample of data that is representative,
and hopefully a close approximation to the population.
Of course, there are many ways we can select our samples,
and there are many books on how to this properly.
One of the simplest ways is to randomly select individual items from the population.
The next obvious question is "how many samples are required?".
The videos coming up will give some ideas on how to answer that.
I will also add, that our computerized data collection systems are so extensive now
that we essentially have the ability to measure the entire population for certain variables.
For example, if we wanted to know the viscosity from our batch reactor,
we can go get this information going back many many years.
This data set, is, in principle, a population.
Statisticians speaking in the strict sense of the definition of "population" will disagree,
but practically-speaking, these many years of data are a population.
Everything that might have occurred to cause variability
in those data are likely represented there: process upsets, summer vs winter effects,
different operators managing the batch, and so on.
In this course you will notice a continual emphasis on the practical use statistics,
rather than sticking to definitions rigidly.
Let's look at the idea next that probability represents the area under a histogram.
If we have scaled our histogram to have a relative frequency axis,
then the area under the histogram is 1, or a 100%.
This allows us to ask interesting questions.
On the plot on the left, what is the probability
that the number thrown on the next dice is a 1 or a 2?
On the plot on the right, what is the probability of having a batch yield greater
than or equal to 170 grams per litre?
It all comes down to the fraction of the area under the histogram.
In the dice example, the bars for 1 and 2 represent about 33% of the area,
so the probability of observing a 1 or 2 is about 33%.
This is an intuitive concept that you are likely familiar with already.
The next terminology we need to consider is the difference between a parameter and a statistic.
A parameter is a value that describes the population.
It is a certain, fixed value; it does not change.
For example, the average board thickness from all boards produced last year is a fixed value.
A statistic is an estimate of the population parameter.
If I randomly select the thickness of one board every hour,
I will have about 8760 board thicknesses.
The average of that sample of data will be an estimate of the population.
So the true average, the parameter, is fixed, but unknown to us.
The estimated average, the statistic, is an estimate of the unknown parameter.
If I went back to my data set and randomly selected a different set of 8760 boards
from last year, I will get a different average, a different statistic.
The parameter has not changed.
Let's now look at the most common statistics: the average, or the mean is the first one.
We say the mean is a measure of location, it is an estimate of where the distribution is,
or we will find the central point.
For a population you would theoretically add up all the individual measurements, the x-values,
and divide by capital N. We denote it with Greek letter "mu".
In practice we will estimate the mean using the samples, the x_i values, and so then we sum
up all the sampled x_i's and divide through by lower case "n".
We denote this sample mean with x-bar.
As described in a prior video on histograms, we also need an estimate of spread.
So here we can use the variance.
The variance for a population will theoretically sum
up over all the values first subtracted from the mean.
These deviations then, some are positive, some are negative, are first squared.
Then we sum these squared values.
Once we've done that, divide by capital N. Now if the variable, x, has units of kilograms,
for example, the variance will have units of kilograms squared.
That's hard to interpret.
So we often use the square root of the variance,
and we call that by a new name: the standard deviation.
The standard deviation has the units of the original variable,
and gives us an idea of the measure of spread.
If we want a statistical estimate of the variance we follow the same approach.
This time we subtract from the sample mean however.
Because the equation defining the mean places a constraint
in our system, it removes a degree of freedom.
So that explains why we divide by "n-1".
You should always check your software: some software will divide by "n",
and others by "n-1", with most doing the latter.
Some people make a big deal out of the denominator;
in large engineering data sets it should almost never make a practical difference
if you use "n" or "n-1".
It does matter though for a small dataset.