Research Design in Occupational Education Copyright 1997. James P. Key. Oklahoma State University Except for those materials which are supplied by different departments of the University (ex. IRB, Thesis Handbook) and references used by permission.

MODULE S1

DESCRIPTIVE STATISTICS

All educators are involved in research and statistics to a degree. For this reason all educators should have a practical understanding of research design. Even if an educator is not involved in conducting research, he or she must be involved in the interpretation of the findings of others to remain dynamic in their teaching.

The primary use of descriptive statistics is to describe information or data through the use of numbers (create number pictures of the information). The characteristics of groups of numbers representing information or data are called descriptive statistics. Descriptive statistics are used to describe groups of numerical data such as test scores, number or hours of instruction, or the number of students enrolled in a particular course.

Descriptive statistics - Numbers which are used to describe information or data or those techniques used to calculate those numbers.

Variable (x) - A measurable characteristic. Individual measurements of a variable are called varieties, observations, or cases.

Population (X) - All subjects or objects possessing some common specified characteristic. The population in a statistical investigation is arbitrarily defined by naming its unique properties.

Parameter - A measurable characteristic of a population. A measurable quantity derived from a population, such as population mean or standard deviation.

Sample - A smaller group of subjects or objects selected from a large group (population).

Statistic - A measure obtained from a sample. It is a measurable quantity derived from a sample, such as the sample mean or standard deviation.

Frequency graph - A picture depicting the number of times an event occurred.

Bar graph or histogram - A frequency graph with number of blocks or length of bar representing the frequency of occurrence.

Frequency polygon - A modification of the bar graph with lines connecting the midpoints of the highest point on each bar.

Frequency curve - A modification of a frequency polygon with the sharp corners rounded. The area under the connecting line of the bar graph, frequency polygon, and frequency curve are equivalent and represent frequency of occurrence.

Mean (µ) or () Arithmetical mean - A number having an intermediate value between several other numbers in a group from which it was derived and of which it expressed the average value. It is the simple average formed by adding the numbers together and dividing by the number of numbers in the group

.

Median - The mid point in a set of ranked numbers.

Mode - The number which occurs most often in a group of numbers.

Range - The difference in the highest score and the lowest score in a set of scores. The range is obtained by subtracting the low score from the high score R = xh - xl.

Variance - The mean of the squared deviations of individual numbers from the mean of the group of numbers ; the square of the standard deviation .

Standard deviation - A measure of the deviation of individual numbers from the mean of the group of numbers. It is the mean or average deviation of those numbers from the mean of the set of numbers .

The use of symbols is a convenient way of expressing parameters or statistics in research work. The following symbols are generally used to express parameters and statistics.

 Parameter Statistic Observation of Variable X x Sum (Sigma) Number of observations N n Mean µ (Mu) Variance (little sigma) s2 Standard Deviation (little sigma) s or SD

Note that for the mean, variance, and standard deviation lower case Greek letters are used to symbolize parameters, while lower case Roman letters are used to symbolize statistics.

In the examples given, the raw scores for the 10 pt. quiz are:

10 9 8 8 7 7 6 6 5 4 2

10 9 8 8 7 6 6 5 5 3

10 9 8 7 7 6 6 5 4 3

Calculating Parameters and Statistics

Suppose ten students made the following scores on a five point quiz. Calculate the measures of central tendency and variation.

 Score x 3 1 -2 4 2 2 -1 1 n = 10 1 2 -1 1 Mode = 3 3 3 0 0 Median = 3 5 3 0 0 = 3 4 3 0 0 Range = 4 3 3 0 0 s2= 1.33 2 4 1 1 s = 1.155 4 4 1 1 3 5 2 4 30 0 12

*Note: n - 1 used for small samples (usually less than 30 numbers).

Measures of Central Tendency

Mode = Number which occurs most often

Median = Middle number - 50% above - 50% below

Mean = Average = M = =

Measures of Variation

Range = Distance from highest to lowest score

Variance = Squared standard deviation = s2 =

Standard deviation = average distance of individual numbers from the mean -

s =

Normal Curve

Percentages

Percentages are used to relate how much a part is of the whole and established a basis for comparing information from groups of unequal sizes.

Example

 If there is a 100 question test and we correctly answer 80 of them, our percentage of correct answers is
 Group A had 36 (50%) students who agreed with the statement while Group B had 25 (75%) students who agreed.

SELF ASSESSMENT

1. The primary use of descriptive statistics is to _____________________

2. Define:

a. Descriptive Statistics

b. Variable

c. Population

d. Parameter

e. Sample

f. Statistic

g. Mean

h. Median

i. Mode

j. Range

k. Variance

3. The frequency graph, bar graph or histogram, frequency polygon, and frequency curve are all means of _____________________illustrating data.

4. Label the following symbol as either a parameter or statistic.

Symbol                       Parameter                                      Statistic

5. Identify by correctly labeling the following graphic illustrations of results of a five point quiz taken by ten students.

a._____________________ b. _____________________

c._____________________ d. _____________________

6. Construct a frequency graph, a histogram, a frequency polygon and a frequency curve to graphically illustrate the following data.

Scores on a ten point quiz taken by fifteen students:

Student Score Student Score Student Score

A              5          F          10          K         8

B              7          G          6            L          6

C              8          H          7          M          8

D              7          I           8          N          8

E              9          J          9          O          9

7. Calculate the measures of central tendency (mean, median and mode) and the measures of variation (range, variance and standard deviation) for the following scores.

x
5
6
7
7
7
8
8
8
8
8
9
9
10
10
10

= Mode =

n = Median =

= Range =

Variance =

Standard Deviation =

8. Out of a normally distributed population, what percentage of the population falls within two standard deviations of the mean?

9. Out of 200 students, 150 agreed with the statement. What percentage of the students agreed?