Basic Statistics, Page 1
Basic Statistics
Introduction
The purpose of this learning module is to introduce you to some of the fundamental definitions and
techniques related to analyzing measurements with statistics.
In all the definitions and examples discussed here, we consider a collection (sample) of measurements of a
steady parameter. E.g., repeated measurements of a temperature, distance, voltage, etc.
Basic Definitions for Data Analysis using Statistics
First some definitions are necessary:
o Population – the entire collection of measurements, not all of which will be analyzed statistically.
o Sample – a subset of the population that is analyzed statistically. A sample consists of n measurements.
o Statistic – a numerical attribute of the sample (e.g., mean, median, standard deviation).
Suppose a population – a series of measurements (or readings) of some variable x is available. Variable x can
be anything that is measurable, such as a length, time, voltage, current, resistance, etc.
Consider a sample of these measurements – some portion of the population that is to be analyzed
statistically. The measurements are x1, x2, x3, ..., xn, where n is the number of measurements in the sample
under consideration. The following represent some of the statistics that can be calculated:
Mean – the sample mean is simply the arithmetic average, as is commonly calculated, i.e.,
1
1n
i
i
x x
n
,
where i is one of the n measurements of the sample.
o We sometimes use the notation xavg instead of x to indicate the average of all x values in the sample,
especially when using Excel since overbars are difficult to add.
o The sample mean, although it is the simplest statistic to calculate, is not always as useful as the sample
median, which is discussed later.
Deviation – the deviation of a measurement is defined as the difference between a particular measurement
and the mean, i.e., for measurement i, ii
dxx .
o When considering a group or sample of measurements, the deviation of one particular measurement is
the same as the precision error or random error of that measurement.
o Deviation is not the same as accuracy error. Recall that accuracy error (inaccuracy) is defined as the
difference between a particular measurement and the true value of the quantity being measured:
(accuracy error = xi – xtrue). Because of bias (systematic) error, xtrue is often not even known, and the mean
is not equal to xtrue if there are bias errors.
Average deviation – to get some feel for how much deviation is represented in the sample, we might first
think of averaging all the deviations to obtain some kind of mean or average deviation. It turns out that the
average of all the deviations is zero! Try it for any set of numbers, and you will convince yourself that this is
true. Why? Because by definition, some of the measurements are smaller than the average, and some are
larger, and the average deviation turns out to be a meaningless and worthless calculation – it is always zero.
Average absolute deviation – a better measure of deviation is the average absolute deviation (also called the
average positive error), defined as the average of the absolute value of each deviation. Mathematically,
1
1n
i
i
dd
n
, where |di| is called the absolute deviation or the positive error.
Sample standard deviation – an even better, and more accepted measure of how much deviation or scatter is
in the data is obtained by calculating the sample standard deviation. For n measurements,
2
2
11
11
nn
ii
ii
dxx
Snn
.
o S is kind of like an average of the deviations, but it is constructed by taking the square root of the average
of the squared deviations, since di can be either positive or negative.
o Notice that the denominator is n – 1, not simply n. It turns out that for small sample size (small n), n – 1
yields a better estimate of the standard deviation than does n itself. (Details are beyond the scope of this
course.) As n gets big, the difference between using n or n – 1 in the denominator becomes negligible.
docsity.com
