I. Some Definitions
1. Data
Measurements collected during a scientific observation.

2. Statistics
A branch of mathematics concerned with describing and interpreting data.
There are actually two functions to statistics:
Descriptive
Serves to organize, summarize, and describe data.
Inferential
Serves to make inferences or generalizations about a total set of individuals or events on the basis of data from a smaller group.
3. Population
A collection of subjects or events that share a common characteristic.

4. Sample
A portion of the population.
That is:

Note population and sample are relative terms. While this class might be a sample from the population of all students at the college, all students at the college could be a sample from the population of all students at public universities in the USA.

5. Parameter
A description of some characteristic of a population.
Is fixed or constant & symbolized with Greek letters.
Note that this term has a different meaning to the lay public and to statisticians.

6. Statistic
A description of some characteristic of a sample.
May vary & typically symbolized with English letters.

In light of these later definitions, we can now be more precise in our definitions of the two functions of statistics:

7. Descriptive Statistics
Serves to describe or summarize the parameters of a population.

8. Inferential Statistics
Serves to infer or generalize about the parameters of a population based on statistics from samples.

9. Random Sampling
A procedure of selecting a sample whereby each member of the population has an equal chance of being chosen into the sample.

10. Stratified Random Sampling
Involves subdividing the population into strata or layers and randomly sampling from each. Consider an example of sampling undergraduate college students.

Population    Sample
Freshman 40% Freshman 40%
Sophomore 30% Sophomore 30%
Junior 20% Junior 20%
Senior 10% Senior 10%

II. Types of Characteristics
1. Constant - Does not vary. Always has the same value.
Exs. pi (π), the number of members of a baseball team on the field.

2. Variable - occurs in differing amounts or kinds.
Exs. IQ, height, eye color.

There are actually two kinds of variables:
1. Qualitative - differs in kind or quality but not amount.
Exs. eye color
2. Quantitative - differs in amount but not in quality.
Ex. IQ

There are two kinds of quantitative variables:
1. Discrete or discontinuous - can only occur in integer (whole number) amounts.
Ex. number of children in a family.
2. Continuous - conceivable or possible in any amount.
Ex. height.

III. Scales of Measurement

Are a set of procedures for assigning numbers to things. Note that the act of measurement discretizes (rounds off) a continuous variable because one can never measure a continuous variable exactly. Ex. I am 5'8" tall. However, if you measured me, I would probably be something like 5'8.21332. . . . inches tall.

There are four scales of measurement that you should be familiar with. The first two are sometimes called nonparametric because they have nothing resembling a zero point.

What follows is a figure showing the 3 temperature scales in order to clarify their similarities and differences, as well as a table that summarizes the properties of the 4 scales of measurement as well as the mathematical statements that one can make with them.
A Comparison of Temperature Scales

Properties & Mathematical Statements of the four Scales of Measurement
Scale Properties Math Statements
Nominal or
Categorical
None =, ≠
Ordinal
or Rank
Magnitude =, ≠, <, >
Interval Magnitude,
Equal Intervals
=, ≠, <, >
=, ≠, <, > of intervals
Ratio Magnitude,
Equal Intervals,
& Absolute 0
=, ≠, <, >
=, ≠, <, > of intervals
A/B=K

IV. Some Mathematical Concepts
1. Real or Exact Limits

Since continuous numbers are rounded, they are only approximate. Thus, the number 33 more precisely falls in the range of 32.5-33.5. More generally, the exact limits of a number equals the number plus and minus (±) 1/2 the unit of measurement. Exs.

Number Unit of meas. 1/2 Unit of meas. Exact Limits
Lower Upper
33
1
.5
32.5 33.5
33.3
.1
.05
33.25 33.35
33.33
.01
.005
33.325 33.335
33.333
.001
.0005
33.3325 33.3335

2. Rounding

We will use the following rules, our goal is to round to 10ths, and examples are provided.

Rule Exs.
Unit
of
meas.
1/2
Unit
Remain
decim.
frac.
Exs.
Rounded
1. If remaining decimal fraction
is less (<) than 1/2 unit of
measurement, drop it.
7.34
.1
.05
.04
7.3
2. If remaining decimal fraction
is greater (>) than 1/2 unit of
measurement, increase the
preceding digit by 1.
7.36
.1
.05
.06
7.4
3. If remaining decimal fraction
equals (=) 1/2 unit of measurement,
increase preceding digit by 1 if it
is odd & drop it if it is even. Hence,
this is called the "Odd/Even Rule".
7.35
7.25
.1
.05
.05
7.4
7.2

Some additional examples to make this concept especially clear follow. Assume that the desired unit of measurement is tenths (.1):

Exs.
Unit
of
meas.
1/2
Unit
Remaining
decim. frac.

Rounded
Rule used
7.34999999
.1
.05
.04999999
7.3
1
7.35000001
.1
.05
.05000001
7.4
2
7.35
.1
.05
.05
7.4
3
7.45
.1
.05
.05
7.4
3

Often folks round up when the remaining decimal fraction equals one half the unit of measurement. The above rules are more precise. Example:

Always up   Our rules
Number Rounded Number Rounded
3.5 4 3.5 4
4.5 5 4.5 4
5.5 6 5.5 6
6.5 7 6.5 6
20 22 sums 20 20

Note that when performing a procedure with intermediate calculations leading to a final result, do not round the intermediate calculations. Just round the final result. In fact, use as many decimals as you can in the intermediate calculations. This will lead to a minimal amount of rounding error. Note also that in statistics the final result should not have a bar over the last decimal to indicate that it repeats (called a Vinculum). Simply round the final result using the rules noted above.

3. Summation Sign (∑)

Consider:

Subject X
1 X1=3
2 X2=8
3 X3=10
4 X4=13
5 X5=17

• Scores on a variable are symbolized with capital letters. Ex. X, Y, Z.

• Each score uses the subject number as the subscript. Ex. X1, X2, X3, . . . , XN (where N=the total number of scores).

• "Any Score" is symbolized Xi (or when the situation is clear, just X).

• The following notation is read "the sum of X from i=2 to 4" and the 2 and 4 are called the "limits of the summation".
In this case, the value equals 8 + 10 + 13.

• Note that:

In other words, when the situation is clear, the notation will be simplified.

• Another example:

• Often we will want to compute the ∑X2 as well as the (∑X)2.
∑X2 is where you square each score and then sum them.
(∑X)2 is where you sum all of the scores and then square the sum.
Consider an example:

Subject X X2
139
2525
3636
∑X = 14 X2 = 70
(∑X)2 = 196
Note: X2(∑X)2

4. Summation Rules (shortcuts):

1. The sum of a constant (c) times a variable (X) equals the constant times the sum of the variable.

2. The sum of a constant (c) taken N times equals N times the constant.

3. The sum of two variables (X + Y) equals the sum of each variable summed.
The following data provide a demonstrate/proof of the summation rules (where c=constant):
Subj. X c cX Y (X+Y)
1 3 2 6 1 4
2 5 2 10 4 9
3 6 2 12 7 13
N=3 X=14 c=6 cX=28 Y=12 (X+Y)=26