Introduction to Statistics

statistics

refers to a set of mathematical procedures for organizing, summarizing, and interpreting information

population

the set of all the individuals of interest in a particular study

sample

set of individuals selected from a population, usually intended to represent the population in a research study

variable

 a characteristic or condition that changes or has different values for different individuals

parameter

a value, usually a numerical value, that describes a population. A parameter is usually derived from measurements of the individuals in the population

statistic

a value, usually a numerical value, that describes a sample. A statistic is usually derived from measurements of the individuals in the sample

Descriptive statistics

statistical procedures used to summarize, organize, and simplify data

Inferential statistics

 consist of techniques that allow us to study samples and then make generalizations about the populations from which they were selected.

Sampling error

naturally occurring discrepancy, or error, that exists between a sample statistic and the corresponding population parameter and fundamental errors

correlational method

two different variables are observed to determine whether there is a relationship between them

• Classifies individuals into categories that do not correspond numerical values (e.g gender - male 0 female 1)

• Limitations: do not provide explanation for the relationship, or demonstrate cause-and-effect relationship

Experimental Method

• one variable is manipulated while another variable is observed and measured

• attempts to control all other variables to prevent them from influencing the results.

• Goal: demonstrate cause-and-effect relationship

• an experiment attempts to show that changing the value of one variable causes changes to occur in the second variable

Experimental Method Characteristics

1. manipulation: one variable is manipulated by changing its value from one level to another

2. Control: researcher must exercise control over the research situation to ensure that other, extraneous variables do not influence the relationship being examined

•  random assignment → to distribute the participant characteristics evenly between the two groups so that neither group is noticeably smarter (or older, or faster) than the other

Experimental Method Categories of variables

1. Participant Variables: characteristics such as age, gender that vary from one individual to another

2. Environmental Variables: These are characteristics of the environment such as lighting, time of day, and weather conditions, must be same for all groups

3. independent variable:manipulated by the researcher, consists of the two (or more) treatment conditions to which subjects are exposed; antecedent conditions that were manipulated prior to observing the dependent variable

4. dependent variable: is observed to assess the effect of the treatment

Experimental Method

• Individuals in control condition do not receive the experimental treatment – either receive no treatment or they receive a neutral, placebo treatment 

• Individuals in the experimental condition do receive the experimental treatment

Nonexperimental Methods

• Nonequivalent Groups - subjects have not been randomly assigned to conditions

  • E.g correlational studies 

• the “independent variable” that is used to create the different groups of scores is often called the quasi-independent variable

Constructs

internal attributes or characteristics that cannot be directly observed but are useful for describing and explaining behavior. 

operational definition

identifies a measurement procedure (a set of operations) for measuring an external behavior and uses the resulting measurements as a definition and a measurement of a hypothetical construct.

operational definition components

• it describes a set of operations for measuring a construct

• defines the construct in terms of the resulting measurements

discrete variable

•  consists of separate, indivisible categories. No values can exist between two neighboring categories

• E.g age, major, gender or whole numbers

continuous variable

• an infinite number of possible values that fall between any two observed values. A continuous variable is divisible into an infinite number of fractional parts

  • E.g weight

Nominal scale

consists of a set of categories that have different names. Measurements on a nominal scale label and categorize observations, but do not make any quantitative distinctions between observations

Ordinal scale

• consists of a set of categories that are organized in an ordered sequence, rank observations in terms of size or magnitude

Interval scale

consists of ordered categories that are all intervals of exactly the same size. Equal differences between numbers on scale reflect equal differences in magnitude; zero point on an interval scale is arbitrary and does not indicate a zero amount of the variable being measured

Ratio scale

• an interval scale with the additional feature of an absolute zero point. With a ratio scale, ratios of numbers do reflect ratios of magnitude

X

to represent scores for a variable

N

the symbol for the number of scores in a population

n

symbol for a number of scores in a sample

Σ

to stand for summation

Definition of frequency distribution

An organized tabulation of the number of individuals located in each category on the scale of measurement

Purpose of frequency distribution

Takes a disorganized set of scores and places them in order from highest to lowest

Two elements of a frequency distribution

1. Set of categories that make up the original measurement scale. 2. Record of the frequency for each category

What do X values represent in a frequency distribution table?

The scale of measurement, not the actual scores

Σf = N

The total number of scores in the distribution

Proportion formula

p = f/N

Percentage formula

p(100) = f/N(100)

How to compute ΣX²

Square each score, then sum

Grouped frequency distribution purpose

To provide a simple, organized view of data by grouping scores into intervals

Ideal number of class intervals

About 10

What should the width of class intervals be?

Simple numbers like 2, 5, 10, or 20

Real limits

The actual boundaries for scores on a continuous scale; e.g. X=40 is 39.5–40.5

Apparent limits

The listed upper and lower boundaries (e.g. 40–49)

Abscissa

X-axis in a graph

Ordinate

Y-axis in a graph

Histogram (interval/ratio data)

Bar is drawn above each X, height = frequency; bars touch each other

Modified histogram

Uses stacked blocks instead of bars, one block = one individual

Polygon

A dot above each score (height = frequency), connected by lines; lines go to X-axis one category beyond data range

Bar graph (nominal/ordinal)

Bars do not touch, emphasize distinct categories; used for nominal and ordinal data

Relative frequency

How often a score occurs in relation to total scores (f/N)

Smooth curve

Used for population distributions; eliminates jagged edges of histogram; idealized view

Symmetrical distribution

Left and right sides are mirror images

Skewed distribution

Scores pile on one side and taper off on the other (positive or negative)

Tail of distribution

Where the scores taper off; defines skew direction

Positive skew

Tail points to the right (positive X-axis)

Negative skew

Tail points to the left (negative X-axis)

Percentile rank

Percentage of individuals with scores at or below a specific value

Percentile

The actual score identified by a percentile rank

Cumulative frequency

Sum of all frequencies up to a given category/class interval

What is interpolation?

A method to estimate an intermediate percentile score between two known values

Steps of interpolation (1)

Find the width of the interval on both axes (X and cumulative

Steps of interpolation (2)

Locate intermediate value’s position as a fraction of the interval width: Fraction = distance from top / width

Steps of interpolation (3)

Apply the same fraction to the other scale: Distance = (fraction)(width)

Steps of interpolation (4)

Use the distance from the top to find the corresponding value on the other scale

Definition of central tendency

A statistical measure to determine a single score that defines the center of a distribution

Goal of central tendency

To find the single score that is most typical or representative of the entire group

Every score contributes to the mean

Yes, unless the score added or removed is exactly equal to the mean

Effect of adding/subtracting a constant to all scores

Same constant is added/subtracted to the mean

Effect of multiplying/dividing scores by a constant

Mean is multiplied/divided by the same constant; commonly used for unit conversions

Population mean formula

μ = ∑X / N

Sample mean formula

M = ∑X / n

Alternative view of the mean

The mean divides the total equally or is the balance point of the distribution

Weighted mean steps

1. Add total scores (∑X) from all groups 2. Sum total sample sizes (n) 3. Divide total ∑X by total n

Why overall mean is not midpoint of sample means

Samples contribute unequally due to different sizes, larger sample carries more weight

Definition of median

The midpoint of a distribution; 50

Does median have a formula?

No, it is not algebraically defined

Median and extreme scores

Median is preferred when extreme values distort the mean

Use of median with undetermined/open-ended values

Preferred because it is unaffected by values that cannot be determined or bounded

Median and ordinal scales

Preferred measure of central tendency for ordinal data

When to use the median

Extreme or skewed distributions, undetermined values, open-ended distributions, ordinal scale data

Definition of mode

The score or category with the greatest frequency in a distribution

Notation for mode

No specific notation

Bimodal distribution

A distribution with two modes

Multimodal distribution

A distribution with more than two modes

Major vs minor mode

In multimodal distributions, the taller peak is the major mode and the shorter is the minor mode

When to use the mode

For nominal data, discrete variables, and supplementary shape descriptions

Why use mode for nominal scales

Because mean/median are not computable for non-numerical categories

Mode and discrete variables

Useful when values must remain whole numbers (e.g. number of children)

Use of mode as supplementary measure

Added to mean or median for quick insight into shape of distribution

Definition of symmetrical distribution

Graph’s right-hand side is a mirror of the left-hand side

Mean and median in symmetrical distribution

They are the same

Definition of skewed distribution

A distribution where mean, median, and mode are located at different points due to asymmetry

Negative skew

Scores pile up on the right and tail tapers to the left

Definition of variability

Quantitative measure of the differences between scores; describes spread or clustering of scores

How variability relates to distribution

It defines the distribution in terms of distance

Variability and representation

Measures how well an individual score represents the entire distribution

Definition of range

Distance between the smallest and largest score

Formula for range (standard)

range = Xmax – Xmin

Formula for range (real limits)

range = URL for Xmax – LRL for Xmin

Formula for range (whole numbers)

range = Xmax – Xmin + 1

Definition of variance

Mean of the squared deviations from the mean; average squared distance from the mean

Definition of deviation

Distance of a score from the mean, calculated as X – μ or X – M