Descriptive and Multivariate Statistics (Chapter 9)

Purpose:  Farmiliarize readers with the principles of descriptive and multivariate statistics so that they are better producers as well as consumers of research and police data.

Goals: 

• Summarize large and small data sets
• Examine the integrity of large and small data sets
• Determine which statistics best portray the data
• Compare more than one variable to others
• Apply descriptive statistics to problem solving and data driven decision-making

Statistics

Science of collecting and organizing data and then drawing conclusions based on data.

Question:        

What are the three types of statistics?

Answer:        

Descriptive, multivariate, and inferential

Descriptive Statistics

Descriptive Statistics summarize large amounts of information in an efficient and easily understood manner.

Multivariate Statistics

Multivariate Statistics allow comparisons among factors by isolating the effects of one factor or variable from others that may distort conclusions.

Inferential Statistics

Measurement

Concept of Levels of Measurement

Four Levels of measurement:

1. Nominal
2. Ordinal
3. Interval
4. Ratio

(Each conveys a different amount of information.)

Nominal Level of Measurement

All categories must be exhaustive (covering all observations that  may exist)

Categories must be mutually exclusive (each observation can only be classified one way) Provides names or labels for distinguishing observations.

Lowest Level of Measurement

(EX – Race/Gender – assigned numbers can not be calculated)

Ordinal Level of Measurement

Categories must be exhaustive and mutually exclusive.

Categories must exhibit a degree of difference which indicates order or ranking

Categories are ordered in some way, but the actual distance between these orderings would not have any meaning.

(EX – Opinion based - good, better, best…)

Interval Level of Measurement

Categories must be exhaustive, mutually exclusive and exhibit a degree of difference.

Assumes that all items on a scale have equal intervals between them.

Logical distances between categories expressed in meaningful intervals.

(EX Temp & IQ)

Ratio Level of Measurement

All Characteristics of Interval

Contains a true ZERO point. (A true zero point allows for measuring the total absence of the concept under measure.)

(Ex. Income, weight, time and age)

Implications regarding Levels of Measurement

• Ratio is the highest level of measurement (Includes all characteristics.)
• Researchers should strive for highest level of measurement possible. Lower levels of measurement can not be converted to higher levels, but higher levels can be converted to lower levels.
• (Most Important) The statistical technique to be applied with determine the level of measurement needed.

Distribution of Data Sets

Process by which large and cumbersome data sets are described in a manner that is easily understood.

Frequency Distribution

Allows for a basic description of the data set and for graphical representation. Allows for more efficient management and analysis of large data sets.

Range

A summary statistic that provides limited information but allows for condensing a frequency distribution. The RANGE is obtained by subtracting the Highest Numerical Value from the Lowest Numerical Value. The RANGE is used to obtain a class interval. It is also a simple measure of variation

Grouped Distribution

Process by which the class interval is used to group a Frequency Distribution.

Note: Although the data have been condensed and the distribution has changed, the nature of the data remains the same.

Question:

What is the purpose of Charts and Graphs?

Answer:

To portray the distribution of data for a quick and meaningful understanding.

Percentages

•

Cumulative Percentages

Question:

What should you do with missing data?

Answer:  In cases containing missing data, you cannot determine whether the missing cases would have fallen into a particular segment or class interval.

Option 1: Include a segment labeled “Missing Cases” and include in the total. (Deflates %)

Option 2: Omit the missing cases altogether. (Inflates %)

In either case document whether missing data has been included or omitted.

Measures of Central Tendency

Mean

Outliers

Median (mdn)

The midpoint or middle score of a distribution. The median is not significantly affected by outliers. In the event that there is an even number of scores in the distribution, rank order the distribution and calculate the average value of the two middle scores.

Mode

The mode indicates the most frequently occurring score(s) or label in a distribution. It is possible to have more then one score or interval tie for most occurring. The mode is primarily used for nominal measurements as it provides limited information and is not subject of further statistical analysis.

Measure of Dispersion

Range, Variance and Standard Deviation

Variance (s²)

Standard Deviation

Standard Deviation measures the average distance that each data item is item is away from the mean of all data items in a distribution. It provides insight on how scores in a distribution compare with each other and allows for comparisons between two distributions.

• Any standard deviation value has no real intuitive meaning;
• Most useful in a comparative sense;
• Comparing the relative values for the standard deviation and the mean indicates how much variation there is in a group of cases, relative to the average.

Skewness

Skewness illustrates the spread of scores weighted to one side of the mean. The are three distinct patterns that may emerge from unimodal distributions: normal, positive, and negative.

Normal Distribution

Positively Skewed

Negatively Skewed

Unimodal distribution of scores weighted to the right generally with the mean being the largest value followed by the median and then the mode. Hump to the right with an extended tail to the left.

Question:

What should you use to represent a skewed distribution?

Answer:

Depends on the nature of the data and the purpose of the research or project.

• If extreme scores are considered random deviation use the MEDIAN.
• If extreme scores are considered as normal use the MEAN.

Rates

Used to standardize some measure for comparative purposes. Calculated by dividing raw numbers by a comparable denominator.

Dichotomous Variable

Proportions

Percent Change

•

A percentage change is a way to express a change in a variable. It represents the relative change between the old and the new value.

Note:  Care should be exercised when comparing two time periods exclusively especially with long periods of time in between.

Multivariate Statistics

Regression Analysis

Regression analysis is a procedure for pattern recognition. Regression attempts to plot a line (trendline) through a given set of data points on a graph. Given a strong enough pattern of association, a regression line suggests possibilities for predicting future values. 

Correlation Analysis