Statistics 101: Understanding Variables, Data, and Descriptive Statistics, Slides of Statistics

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Science of Statistics

• Descriptive Statistics
  • methods of summarizing or describing a set of data tables, graphs, numerical summaries
• Inferential Statistics
  • methods of making inference about a population based on the information in a sample

Variables

• Individuals are the objects described by a

set of data; may be people, animals or

things

• Variable is any characteristic of an

individual

Statistical Data

• What purpose do the data have?
• Individuals – Describe? How many?
• Variables – How many? Definition? Unit of measurement?

Types of Variables

• Categorical variable places an individual into

one of several groups or categories

• Quantitative variable takes numerical values

for which arithmetic operations make sense

• Distribution of a variable tells us what values

it takes and how often it takes these values

Exploratory Data Analysis

• Examine each variable by itself… then

relationships among the variables

• Start with graphs… then add numerical

summaries of specific aspects of the data

Levels of Measurement

• Nominal
• Ordinal
• Interval
• Ratio

It's important to recognize that there is a hierarchy implied in the level of measurement idea. At each level up the hierarchy, the current level includes all of the qualities of the one below it and adds something new. In general, it is desirable to have a higher level of measurement.

In nominal measurement the numerical values just "name" the attribute uniquely. No ordering of the cases is implied.

For example, jersey numbers in basketball are measures at the nominal level. Is a player with number 30 more of anything than a player with number 15?

In ordinal measurement the attributes can be rank- ordered. Here, distances between attributes do not have any meaning.

For example, on a survey you might code Educational Attainment as 0=less than H.S.; 1=some H.S.; 2=H.S. degree; 3=some college; 4=college degree; 5=post college. In this measure, higher numbers mean more education. But is distance from 0 to 1 same as 3 to 4?

In interval measurement the distance between attributes does have meaning.

For example, when we measure temperature (in Fahrenheit), the distance from 30-40 is same as distance from 70-80. The interval between values is interpretable. Because of this, it makes sense to compute an average of an interval variable, where it doesn't make sense to do so for ordinal scales. Do ratios make sense at this level? For example, is it twice as hot at 80 degrees as it is at 40 degrees?

Finally, in ratio measurement there is always an absolute zero that is meaningful. This means that you can construct a meaningful ratio.

Weight is a ratio variable. In applied social research most "count" variables are ratio. Is number of clients in past six months ratio? Why?

Describing Graphically

• Bar Graph: count or percent
• Pie Chart: parts of the whole
• Stem Plot: shape of distribution
• Histogram: great when lots of groups
  • Frequency Table

Time Plots

• Time Series: measurements of a variable

taken at regular intervals over time

• Residual Plots: checking assumptions
• Trends, such as seasonal variation

Outliers

• ‘Extreme’ Values
• What do you do with outliers? - Ignore them - Throw them out -?

Graphical Examples

Let’s Take a Look

Choosing a Summary

The five-number summary is usually

better than the mean and standard

deviation for describing a skewed

distribution or a distribution with

strong outliers.

Use the mean and standard deviation

for reasonably symmetric distributions

that are free of outliers.

Describing Distributions with Numbers

• Mean: simple average
  • is sensitive to extreme scores
  • not necessarily a possible value
• To calculate: add the values and divide by the

number of items

• Median: middle score
  • not sensitive to extreme scores
• To Calculate:
  • rank data from smallest to largest
  • if n is odd, median is the middle score
  • if n is even, median is the average of two middle scores

• Mode: most frequent score
  • does not always exist
  • unstable
  • can be used with qualitative data

Measures of Dispersion (Variability)

• Range
  • totally sensitive to extreme scores
  • easy to compute
• To Calculate: high score – low score

• Variance: measures squared distances

from the mean

• large values of suggest large variability
• Standard Deviation : square root of the

variance

Empirical Rule

• Should be used for ‘mound shape’ data
  • approx. 68% of the data fall between mean +/- SD
  • approx. 95% of the data fall between mean +/- 2 * SD
  • approx. 99.7% of the data fall between mean +/- 3 * SD

Let’s give it a try

• Let’s use faculty experience.
• Why?
• What should we do with it?

Quartiles and 5-Number Summary

• Quartiles divide ordered numerical data into four equally sized parts. - 1 st^ quartile, Q1, 25% below and 75% above - 2 nd^ quartile, Q2, median, 50% below and 50% above - 3 rd^ quartile, Q3, 75% below and 25% above
• The low score, Q1, Q2, Q3, and the high score are known as the five number summary of a data set.

BoxPlots

• Particularly helpful in comparing 2 or more groups
• Box shows central 50% of data and the median
• Whiskers show extremes

Let’s give it a try

• Let’s use the $ in the pocket data.
• Why?
• What should we do with it?