Preliminary Concepts
 Some Definitions
 Types of Characteristics
 Scales of Measurement
 Some Mathematical Concepts
 Real or Exact Limits
 Rounding
 Summation Sign
 Summation Rules
Practice Problems (Answers)
Homework
I. Some Definitions
 Data
 Measurements collected during a scientific observation.
 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.
 Population
 A collection of subjects or events that share a common characteristic.
 Sample
 A portion of the population. (Note population & sample are relative terms).
That is:
Parameter
 A description of some characteristic of a population.
Is fixed or constant & symbolized with Greek letters.
 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:

 Descriptive Statistics
 Serves to describe or summarize the parameters of a population.
 Inferential Statistics
 Serves to infer or generalize about the parameters of a population based on statistics from samples.
 Random Sampling
 A procedure of selecting a sample whereby each member of the population has an equal chance of being chosen into the sample.
 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
 Constant  Does not vary. Always has the same value.
Exs. pi (π), the number of members of a baseball team on the field.
 Variable  occurs in differing amounts or kinds.
Exs. IQ, height, eye color.
There are actually two kinds of variables:
 Qualitative  differs in kind or quality but not amount.
Exs. eye color
 Quantitative  differs in amount but not in quality.
Ex. IQ
There are two kinds of quantitative variables:
 Discrete or discontinuous  can only occur in integer (whole number) amounts.
Ex. number of children in a family.
 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.
 Nominal or Categorical
Used to distinguish different categories for qualitative variables. Gives the same number to members of the same category and different numbers to members of different categories. In other words, the numbers here are essentially "dummy codes."
Exs. Gender, ethnic background.
 Ordinal or Rank
Uses numbers in a manner such that the numbers are in the same relationship as the characteristic is among the different members of the group of people or things. In this case, the numbers indicate position in an ordered series and not how much of a difference exists between the successive positions on the scale.
Exs. Hardness of rocks, Beauty.
 Interval
Is a parametric scale (i.e., it has a fixed unit of measurement) and has an arbitrary zero point (which means the zero point does not truly reflect absence of the characteristic).
Exs. Celsius or Fahrenheit temperature.
 Ratio
Is also a metric scale because it has an absolute zero (which truly reflects absence of the characteristic).
Exs. Kelvin temperature, speed, height.
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
 Real or Exact Limits
Since continuous numbers are rounded, they are only approximate. Thus,
the number 33 more precisely falls in the range of 32.533.5. 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 
 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.
 Summation Sign (∑)
Consider:
Subject 
X 
1 
X_{1}=3 
2 
X_{2}=8 
3 
X_{3}=10 
4 
X_{4}=13 
5 
X_{5}=17 
 Scores on a variable are symbolized with capital letters. Ex. X, Y, Z.
 Each score uses the subject number as the subscript. Ex. X_{1}, X_{2}, X_{3}, . . . , X_{N} (where N=the total number of scores).
 "Any Score" is symbolized X_{i} (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 ∑X^{2} as well as the (∑X)^{2}.
∑X^{2} 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 
X^{2} 
1  3  9 
2  5  25 
3  6  36 

∑X = 14 
∑X^{2} = 70 
(∑X)^{2} = 196 
Note: ∑X^{2}
≠ (∑X)^{2} 
 Summation Rules (shortcuts):
 The sum of a constant (c) times a variable (X) equals the constant
times the sum of the variable.
 The sum of a constant (c) taken N times equals N times the constant.
 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 
Copyright © 19972016 M. Plonsky, Ph.D.
Comments? mplonsky@uwsp.edu.