Test 1 - Statistics

Statistics

Statistics is the science of gathering, organizing, analyzing and drawing conclusions from numerical data

Units of Statistical Analysis

Any entities that our data describe (aka units of observation, or cases)

– Individual people

– Schools, universities, organizations

– Geographical areas

– Countries

– Entire world at different points in time

Variables

• We characterize each unit of analysis by a number of traits and attributes = variables

• Variable is any characteristic that can have different values (at least 2)

• Examples: race, income, education — for individuals; Gross Domestic Product (GDP), literacy rate, infant mortality rate — for countries

Note: If everyone has the same value on a characteristic, it’s a constant. For example, in a dataset where units of statistical analysis are college students, being a college student is a not a variable but a constant.

Nominal

• Values vary in quality but not the amount

• Examples: nationality, religion, occupation

• Can be represented by numbers, but still not

quantitative (e.g., student ID values)

• Categories should be exhaustive (nothing left

out) and exclusive (no overlap)

• Dichotomies (yes/no variables) are always

nominal

Ordinal

• Numbers denote order, ascending or

descending

• Distances between numbers not defined

• Examples: agreement scales, approval scales, pain scales (when these are based on one single question)

Interval (Continuous)

• Distances between numbers are meaningful

• Difference between 1 and 2 is the same as

between 10 and 11

• Example: Fahrenheit temperature scale

Ration

Ratio: has a natural zero point (= total lack of)

• Examples: age, weight, income

• That 0 value may never occur in the data (e.g., one can’t weigh 0 lbs

Interval + Ratio

Interval and ratio variables are analyzed and treated the same in statistics

Jointly, they are called continuous or scale

Level of Measurement Decision Tree

Frequency Distributions

- Frequency distribution → always a first step for

nominal and ordinal variables; can be used for

interval/ratio if not too many distinct values

• Frequency distribution = the only way to show

variability/spread for nominal variables

• Cumulative frequency distribution → only for

ordinal and interval/ratio, mostly useful for

technical reasons: to find median and IQR,

identify outliers (e.g., top/bottom 1% or 5%)

Central Tendency: When to use?

• Nominal → mode only, NEVER mean or

median

• Ordinal → median or mode; mean is

sometimes used but that’s not technically

correct

• Interval/ratio → mean (if no outliers) or

median (if outliers or skew); mode can be

used but often not as useful, especially if

there are many values, unless distribution is

bimodal or multimodal

Variablity/Spread: When to Use?

Nominal → entire frequency distribution

• Ordinal → entire frequency distribution, range

& IQR (and even though technically incorrect,

some use mean + SD/variance)

• Interval/ratio → can use all of them, most

commonly = SD and IQR (variance is used for

more technical reasons, to be demonstrated

later)

• Media rarely present measures of variability

What to Use to Describe a Single Variable?

Box-and-Whisker Plot (Boxplot)

Normal Curve (Bell Curve)

• The most well known “shape” is the normal

curve

• Random processes (chance) often result in a

normal curve

Measures of Distribution’s Shape: Kurtosis

Measures of Distribution’s Shape: Skewness

Interpretation of Skewness and Kurtosis Statistics

Skewness:

– Near zero (-0.5 to 0.5) = symmetric (close to normal)

– Positive = right skew (.5 to 1 = moderate, >1 high)

– Negative = left skew (-.5 to -1 = moderate, <-1 high)

Kurtosis:

– In Stata: subtract 3 before interpreting!

– Near zero (-0.5 to 0.5) = close to normal

– Positive = leptokurtic (.5 to 1 = moderate, >1 high)

– Negative = platykurtic (-.5 to -1 = moderate, <-1 high)

Univariate vs Bivariate Bar Graph

• Univariate Bar Graph = percentages for

categories of one nominal/ordinal variable →

add up to 100%

• Bivariate Bar Graph = represents the mean or

percentage of something calculated separately

for each group (2 variables – one main

variable and one group variable) → does not

add up to 100%

Histogram or Univariate Bar Graph?

• Bar graph (nominal/ordinal variable or 2 vars):

– Spaces between bars

– Bars separately labeled

• Histogram (one interval/ratio variable):

– No spaces between bars

– No individual labels for bars

Bar Graphs Should:

Follow the proportional ink principle

• Always include 0 on vertical axis

• Include proper labels on both axes (vertical

axis label can be skipped if in the title)

• Include values for each bar

• Space the bars equally

• Do not leave any relevant data out

Line Instead of Bars = Line Plot

(Always Bivariate)

Line Graphs Should:

• Include proper labels on both axes

• Have equal spacing on both axes (no skipping)

• Not leave any relevant data out

• Have appropriate vertical scale (not too big or

too small)

• Do not need 0 on vertical axis (unless the area

is shaded)

Pie Charts Should

• Have slices add up to 100% (one variable only)

• Display mutually exclusive and exhaustive

categories

• Follow proportional ink principle (no 3D)

• Label categories and list percentages

Proportional Ink Principle Violations:

3D Charts