Preliminary Concepts

1. Some Definitions
2. Types of Characteristics
3. Scales of Measurement
4. Some Mathematical Concepts
   1. Real or Exact Limits
   2. Rounding
   3. Summation Sign
   4. Summation Rules

Practice Problems (Answers)
Homework

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%

1. Constant - Does not vary. Always has the same value.
   Exs. pi (p), 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.

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.

1. 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.
2. 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.
3. 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.
4. Ratio
   Is also a metric scale because it has an absolute zero (which truly reflects absence of the characteristic).
   Exs. Kelvin temperature, speed, height.

A Comparison of Temperature Scales

Properties & Mathematical Statements of the four Scales of Measurement

Scale	Properties	Math Statements
Nominal or Categorical	None	=, ¹
Ordinal or Rank	Magnitude	=, ¹, <, >
Interval	Magnitude, Equal Intervals	=, ¹, <, > =, ¹, <, > of intervals
Ratio	Magnitude, Equal Intervals, & Absolute 0	=, ¹, <, > =, ¹, <, > of intervals A/B=K

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.
   
   
3. Summation Sign (å)

   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, . . . , 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 åX² as well as the (åX)².
     åX² is where you square each score and then sum them.
     (åX)² is where you sum all of the scores and then square the sum.
     Consider an example:
     
     

     Subject	X	X2
     1	3	9
     2	5	25
     3	6	36
     	åX = 14	åX2 = 70
     	(åX)2 = 196
     Note: åX2 ¹ (åX)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	åX=14	åc=6	åcX=28	åY=12	å(X+Y)=26