2.7 - Connecting measures together

Connecting Measures of Central Tendency & Measures of Dispersion

Previous sections introduced two classes of measures in descriptive statistics:

• central tendency
• dispersion
  Those class separately don't give us much insight onto the whole data, but put together, we can get good understanding of the data set.

Example from 2.6

Now, let's as a question - What % of observations fall within at most 1 standard deviation from the mean?

For 1 stdev:
$\mu -1\sigma =26.125-15.6=10.525$
$\mu +1\sigma =26.125+15.6=41.725$
So the values falling in $[\mu -\sigma ,\mu +\sigma ]$ are $\{13,21,28,33\}$

Cherbyshev's Inequality

Using that, we can guarantee a minimum percentage of observations falling in an interal symmetric about the mean
If $k=2$, we have at least $1-\frac{1}{s^2}=\frac{3}{4}=75\mathrm{\%}$ of the observations fall between $\mu -2\sigma$ and $\mu +2\sigma$

Half of all observations fall for $k=\sqrt{2}$, as $1-\frac{1}{{\sqrt{2}}^2}=1-\frac{1}{2}=\frac{1}{2}$

Normal distribution and Curve fitting

Cherbyshev's Inequality applies to all distributions, but more precise connections can be made when we restrict our interest to particular classes of distributions
To express the shape of a normal distribution, we must discuss modelling distributions with functions or curves. Consider the following

The curve highlights the shape of histogram. Increasing the number of classes would make the curve be better suited, but might not be possible with discrete variables and finite data sets.
The bar height represents the percentage of observations in that class
To easily read the value of function, we fit a curve that closely resembles common classes.

For bell curve:

• Uniquely devined by mean and standard deviation
• Inflection points are one standard deviation away from the mean
• mean = median = mode
• Symmatric about $\mu$
• Are under curve over an interval is the relative frequency for the interval

Inflection point of standard bell curve is exactly at one $\sigma$ away from $\mu$

Empirical Rule for Normal Distributions

Implication of the Empirical Rule

Suppose the IQ distribution has mean of 100 and stdev of 15

Determine % of adults with IQ between 70 and 130
70 is $\mu -2\sigma$ and 130 is $\mu +2\sigma$, so its $95\mathrm{\%}$
Determine % of adults with IQ greater than 100
100 is the mean, so $50\mathrm{\%}$
Determine % of adults with IQ less than 55
$\frac{55-100}{15}=-3$ so its $3\sigma$ below the mean, which means $0.15\mathrm{\%}$
Determine % of adults with IQ between 85 and 145
<85 is 1 deviation, >145 is 3 above
1 below is 34%, 3 above is 49.85% which in total is 83.85%
Determine % of adults with IQ between 95 and 105
$\frac{95-100}{15}=\frac{1}{3}=\frac{105-100}{15}$, so its $\frac{1}{3}\sigma$ away from mean
we cannot calculate that right now, but we'll be able to in ch.4

z-score

z-score of an observation is how many standard deviations away from the mean it is
$z>0$ - observation is greater than the mean
$z<0$ - observation is less than the mean

Examples

$z=0$ - equal to the mean
$z<-2$ - below the $2^{nd}$ deviation

Student got $76\mathrm{\%}$ on an exam. Class average was $71\mathrm{\%}$ and stdev $2.5\mathrm{\%}$
$z=\frac{x-\mu }{\sigma }=\frac{76-71}{2.5}=\frac{5}{2.5}=2$
Student got $85\mathrm{\%}$ on an exam. Class average was $80\mathrm{\%}$ and stdev $3\mathrm{\%}$
$z=\frac{x-\mu }{\sigma }=\frac{85-80}{3}=\frac{5}{3}$

We cannot compare the scores against the other students grades, but we can say that the first exam was taken much better than the first, against the averages of the class

Classifying Observations Comparatively

Different errors may occur, and some are more egregious than others.
Various protocols can find outliers - values far away from the majority of data values
Unusual observation value is top-most or bottom-most percentiles of the data set

These characterizations don't provide standard for determining classification, just a distinction between ideas

Definitions