Now, for reasons I don't want to get into because it's rather technical and

we'll need some more advanced statistical theory to address this.

When we take an average of these squared deviations from the mean,

we don't actually divide by n.

We divide by n-1.

Now for the purposes of this recording and this course, just trust me on this fact.

But if you decide to take your studies of statistics to a much greater level

Will discover the reason for why.

But for practical purpose if n is very large, whether one divides by n or n or

n minus 1 makes very little practical difference.

So in summary the sample variance is an average, the average squared

deviation about the mean, just with this divisor of n minus 1 rather than n.

So as a dataset has more variability within it, if we return to those share

returns, the red stock had much greater variation than the black stock.

Then the average of those square deviations about the mean, ie the sample

variance, would be much greater for the red stock over the black one.

So, we have the sample variance S squared.

Now, if our variable was measured in particular units,

then as we have these squared deviations about the mean

in our variance calculation, then the actual units of measurement

of the variance would always be the square of the original units.

Now frequently, the square of the original units may have no practical meaning.

So if we're looking at share returns, let's say as percentage returns,

the percentage on a given day.

Then the units of the variants would be percent squared.

Now that's a concept many of us can't easily relate to.

Of course, sometimes the square of a unit may have some real world meaning,

if we were looking at height, for example, measured in metres.

Then the variance of height would be measured in metres squared, well,

you can also roughly visualise what a square metre looks like.

But in general, the squared unit has little real world applicability.

Which is why in practice we tend not to focus so much on the sample variance

s squared, rather its positive square root, the sample standard deviation.

Because if we square root the variance,

that means we are square rooting those squared units of measurement, and

hence we return to the original units of measurement.

So, it's very common to deal with the standard deviations over the variances,

just to ease the interpretation of the result.

So for any of you, perhaps they're interested in finance, and you'd like to

know what are the key measures of risks or volatility in the financial markets.

Be it related to the returns on an individual stock or

maybe movements in an exchange rate.

Well, the standard deviation is a very widely used measure in finance.

The statistical measure to simply try and

quantify the risk associated with any particular asset.

So if we sort of bring together our measures of central tendency, and

combine it with measures of dispersion,

we can now focus on two key features of attributes of a distribution.

Some sense about where it's centered, what is the typical perhaps return,

the average return say on a stock.

Typically denoted by our sample mean x bar,

though conscious of its sensitivity to any outliers in our dataset.

But also some measure of the dispersion and in terms of financial assets.

The risk of holding these assets,

which we could quantify either through all the sample variance.

But to ease the interpretability,

will tend to focus on the sample standard deviation instead.

So these are some very widely used and very important descriptive statistics, so

do make sure you're comfortable with these, and look out for

these as when they're mentioned in the media.

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