measurement and descriptive statistics

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CHAPTER 3Measurement and Descriptive Statistics

Outline

3.1. Frequency distributions3.2. Levels of measurement

3.2.1. Nominal3.2.2. Ordinal3.2.3. Interval3.2.4. Ratio

3.3. Descriptive Statistics and Plots3.3.1. Frequency tables3.3.2. Bar charts3.3.3. Histograms3.3.4. Frequency Polygons3.3.5. Box and Whiskers Plot3.3.6. Measures of Central Tendency3.3.7. Measures of Variability3.3.8. Standard Deviation3.3.9. Interquartile range3.3.10.Measurement and Descriptive Statistics

3.1. Frequency Distributions

A frequency distribution is a tally or count of the number of times each score on a single variable occurs.

Grade Frequency

A 7

B 20

C 18

D 5

Total 50

Example:

Approximately normal distribution

The largest number of students has scores between 450 and 550 (the middle two bars).

Small numbers of students have very low and very high scores.

Approximately normal distribution

When there are most scores for the middle values and a small number of scores for the low and high values, the distribution is said to be approximately normal.

normal, bell shaped curve

Non-normal distribution

The tail of the curve is on the low end or left side.

This distribution can be said to be NOT normally distributed.

different from normal curve

3.2. Levels of measurement

Nominal

Ordinal

Interval

Ratio

3.2.1. Nominal

Two or more unordered categories

Example:

3.2.2. Ordinal

Ordered levels, in which the difference in magnitude between levels is not equal.

Example:

very dissatisfied

somewhat dissatisfied

somewhat satisfied

1

2

3

4very satisfied

3.2.3. Interval

Ordered levels, in which the difference in magnitude between levels is equal

Example: Temperature, Dates (data that has an arbitrary zero), etc.

3.2.4. Ratio

Interval data with a natural zero point (data that has an absolute zero)

Example: height, weight, age, length, ruler, year of experience, etc.

Identify the scale of measurement for the

following: military title -- Lieutenant,

Captain, Major.

A. nominal

B. ordinal

C. interval

D. ratio

Question 1

Question 2

Identify the scale of measurement for the

following categorization of clothing: hat, shirt,

shoes, pants

A. nominal

B. ordinal

C. interval

D. ratio

Question 3

Identify the scale of measurement for the

following: heat measured in degrees

centigrade.

A. nominal

B. ordinal

C. interval

D. ratio

Question 4

A score on a 5-point quiz measuring knowledge

of algebra is an example of a(n)

A. nominal

B. ordinal

C. interval

D. ratio

Question 5

Amount of money you have in your pocket

right now (25 cents, 55 cents, etc.) is an

example of:

A. nominal

B. ordinal

C. interval

D. ratio

3.3. Descriptive Statistics and Plots

Frequency tables

Bar charts

Histograms

Frequency Polygons

Box and Whiskers Plot

Measures of Central Tendency

Measures of Variability

Standard Deviation

Interquartile range

Measurement and Descriptive Statistics

3.3.1. Frequency Tables

A frequency table is a table that shows the total for each category or group of data.

3.3.2. Bar chart

Bar charts are used well for the frequency distribution of variables like religion, ethic group or other nominal variables.

0

5

10

15

20

25

30

35

Protestant Catholic No religion

3.3.3. Histograms

Histograms look like bar charts; however there is no space between the boxes, indicating that there is a continuous variable theoretically underlying the scores.

3.3.4. Frequency Polygons

It connects the points between the categories and is best used with approximately normal data. It is also used with ordinal data.

3.3.5. Box and Whiskers Plot

It is a graphical representation of the distributions of scores and is helpful in distinguishing between ordinal and normally distributed data.

3.3.6. Measures of Central Tendency

Three measures of the center of a distribution are commonly used: mean, median and mode.

1. Mean: sum of the values divided by the number of cases

Example: We have a set of data: 15, 24, 49, 8, 50. Mean: (15 +24 + 49 +8 +50)/5 = 29. 2

2. Median: the middle score or median is the appropriate measure of central tendency for ordinal level raw data.

The median represents the middle of the ordered sample data- When the sample size is even, the median is the midpoint/mean of the two middle values

- When the sample size is odd, the median is the middle value.

3. Mode: is the number that appeared frequently in the data set.

3.3.7. Measures of Variability

Variability tells us about the spread or dispersion of the scores. 1. Range: the area of variation between upper and lower

limits on a particular scale.

Example: 8, 15, 24, 47, 50

=> Range = 50 – 8 = 42.

2. Standard deviation: (SD) measures the amount of variation or dispersion from the average.

Example: 1, 2, 3, 4, 5+ Mean: = (1 +2+3+4+5)/ 5 = 3+ (X- )² : (1-3)² = 4, continue to do like this and then we have the sum of this is 10. + Apply the formula. The result is: 1.58

SD

3.3.8. Interquartile range

In descriptive statistics, the interquartilerange (IQR), is a measure of statistical dispersion,being equal to the difference between the upperand lower quartiles, IQR = Q3 − Q1

Example 1: if the number of values is odd

Example 2: if the number of values is even

3.3.9. Measurement and Descriptive Statistics

- Statistics based on means and standard deviation are valid for normally distributed or normal data.

- Typically, these data are used in the most powerful tests called - parametric statistics. However, if the data are ordered but grossly non–normal, means and standard deviations may not give meaningful answers. Then the median and a nonparametric test would be preferred.

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