Frequency distribution

ORGANIZING AND ANALYZING MEASUREMENT FOR EVALUATION
Purpose of statistics: To describe a group or the characteristics of a group concisely and precisely.
Descriptive: The methods used to describe the group are called descriptive statistics. The area of descriptive statistics involves the description of a group (set) of individuals or items by means of graphic, tabular, or numerical devices. For example, determining:

Frequency - the center and spread of the distribution can be calculated
Typical or usual value
Amount of variability
Degree of relationship
The reliability and validity of an instrument

MEASUREMENT SCALES
A process of assigning numbers to objects or events.
Different ways of treating numbers.
Each type of scales represents a way of assigning numbers.

The interpretation of the results depends on the type of scale:

Nominal: use to identify an object or person. Naming or classifying, such as football position, gender, or type of car. A nominal scale is categorical in nature, simply identifying differences among things on some characteristic. The is no notion of order, magnitude, or size.
Ordinal: ranking such as the finishing place in a race. “things” are ranked in order, but the differences between ranked positions are not comparable (i.e., the difference between rank number 1 and 2 may be quite small, but the difference between rank number 4 or 5 may be very large)
Interval: using an equal or common unit of measurement, such as temperature F or IQ. The zero point is arbitrary chosen. That is, a value of zero simply represents a point on a number line. It does not mean that something does not exit.
Ratio: possessing an absolute (true) zero, such as weight or height, salary, etc.

Characteristics Measurement Scales

Scale	Properties
Nominal	Indicates a difference
Ordinal	Indicates a difference Indicates the direction of the difference (e.g. more than or less than)
Interval	Indicates a difference Indicates the direction of the difference Indicates the amount of the difference (in equal intervals)
Ratio	Indicates a difference Indicates the direction of the difference Indicates the amount of the difference (in equal intervals) Indicates an absolute zero

CONTINUOUS VS DISCRETE VARIABLES
Continuous: precision of measurement. can be measured to continuing finer degrees (distance, time, weight, mass)
Discrete: to measure frequency or counting type. How many?

FREQUENCY DISTRIBUTION
Scores representing subscapular skinfold thickness for 9 year-old girls:

Data set of 10 scores (in mm): 6,9,7,10,2,4,9,5,3,6
10-9-9-7-6-6-5-4-3-2

In a small set of scores, interval size of 1 is good

Score   Tally   f       cf      c%
10      x       1       24      100.0
9       x       1       23      95.8
8       xx      2       22      91.7
7       xxxx    4       20      83.3
6       xxxxxx  6       16      66.7
5       xxxxx   5       10      41.7
4       xxx     3       5       20.8
3       x       1       2       8.3
2               0       1       4.2
1       x       1       1       4.2

tally represents the recording of a score in the appropriate interval
f consists of the frequency of scores in each interval
cf represents the cumulative frequency. The numbers are obtained by summing the frequencies from the bottom interval to the top (cf for interval 6 is 16 - 16 out 24 people obtained scores of 6 or below)
c % - cumulative percent, represents the conversion of the cf to percentages. (divide cf by total number)
c% = cf/N * 100
c% = 16/24 * 100 = 66.7%
c% = 66.7
For example, in the above frequency distribution, in Interval 6 66.7% of the group of 24 obtained scores of 6 or below. This column is similar to percentiles.
Interval size
Larger set of data used larger intervals
General rule - use no fewer than 10, no larger than 20
Formula
(HS - LS + 1)
80 - 21 + 1 = 60

FREQUENCY DISTRIBUTIONS

Occasionally we may need to reduce a frequency distribution into a more compact picture. To do this, we group our scores into intervals.

Look at patterns


Scores  f               Scores  f
90-99   8               95-99   2
80-99   20              90-94   6
70-79   14              85-89   
60-69   8               80-84   
                        75-79   
                        70-74   
                        65-69

Better more intervals 10-15
Score limits: upper and lower limits of intervals in a frequency distribution represented in raw score units, i.e., the actual scores obtained on the test.
Real limits: is 1/2 unit above the apparent upper limit and 1/2 unit below the apparent lower limit.
Midpoint: The exact center of any interval

Percentile rank: tells a person his or her relative standing in a class or group. A person who did 35 push-ups performed as well or better than the 91% of the his classmates.

CONSTRUCTING A FREQUENCY DISTRIBUTION

1) Locate the highest score and lower score
Example: 

        Highest score = 10
        Lower score =     1

2) Calculate the range (highest score - lower score + 1)
Example:

        R = highest score - lower score + 1

3) List all possible scores in the range.  The highest score should be placed at the top of the list.
Example:

Score

10 9 8 7 6 5 4 3 2 1


4) Tally each of the scores in the data set using x’s or |’s
Example:

Score   Tally
10      x
9       x
8       xx
7       xxxx
6       xxxxxx
5       xxxxx
4       xxx
3       x
2       
1       x

5) Develop the frequency (f) column by summing the tallies for each interval
Example:

Score  Tally   f
10      x       1
9       x       1
8       xx      2
7       xxxx    4
6       xxxxxx  6
5       xxxxx   5
4       xxx     3
3       x       1
2               0
1       x       1

6) Generate the cumulative frequency (cf) column by adding the frequencies , beginning with the 
bottom interval.
Example:

Score   Tally   f       cf
10      x       1       24
9       x       1       23
8       xx      2       22
7       xxxx    4       20
6       xxxxxx  6       167
5       xxxxx   5       10
4       xxx     3       5
3       x       1       2
2               0       1
1       x       1       1



7) Develop the cumulative percent (c%) column by dividing cf by N and multiplying by 100
Example:

Score   Tally   f       cf      c%
10      x       1       24      100.0
9       x       1       23      95.8
8       xx      2       22      91.7
7       xxxx    4       20      83.3
6       xxxxxx  6       16      66.7
5       xxxxx   5       10      41.7
4       xxx     3       5       20.8
3       x       1       2       8.3
2               0       1       4.2
1       x       1       1       4.2



If the frequency distribution includes intervals larger than size 1, two additional steps, 



8) Select an appropriate number of intervals to ensure that the distribution will contain no more that 
20 intervals and no fewer than 10.


highest score= 99

lowest score =  80



Range = 99 - 80 + 1 = 20

i= 10



9) Select an appropriate interval size.  Remember that typical interval sizes are 2, 3, 5, 7,. and 10.  To 
determine the interval size, the number of intervals can be divided into the range.  

R/i = interval size

20/10 = 2

98-99
96-97
94-95
92-93
90-91
88-89
86-87
84-85
82-83
80-81



GRAPHING DISTRIBUTIONS
Picture the entire distribution
Histogram

1) Hight of graph 3/4 of the width
    draw the X-axis - scores
2) Draw a vertical line (y-axis) at the left end of the x-axis
   - frequency
3) Draw lines parallel to the y-axis to the hight of the  frequency
4) Give a title

Frequency Polygon

1) Hight of graph 3/4 of the width
draw the X-axis - scores
2) Draw a vertical line (y-axis) at the left end of the x-axis - frequency
3) Draw points above midpoint of each interval.