(5) STATISTICS

Measurement

the act of assigning numbers or symbols to characteristics of things

Error

collective influence of all the factors on a test score or measurement beyond those specifically measured by the test or measurement

Scales

set of numbers or symbols where properties empirical properties of the objects to which numbers are assigned

Continuous Scale

countable set of values (can be infinite)

Discrete Scale

countable in a finite amount of time

Magnitude

The properties of moreness and comparison

Equal Interval

The differences between two variables at any place on the scale has the same meaning as the difference between two other points that differ by the same number of scale unit

Ratio

Obtained when nothing of the property being measured exists

Nominal Scale

categorized using names, labels, or qualities

Nominal Scale

• Gender

• Eye color

• Types of fruit

• Political affiliation

Ordinal Scale

can be categorized, and the categories have a meaningful order or rank

Ordinal Scale

• Educational levels

• Satisfaction ratings

• Likert scales

• Race rankings

Interval Scale

have all the properties of ordinal data, plus the intervals between consecutive values are equal and meaningful; does not have absolute zero point

Interval Scale

• Temperature in Celsius or Fahrenheit

• IQ scores

• Calendar years

Ratio

have all the properties of interval data, plus a true or absolute zero point

Ratio

• Height, weight, and length

• Age

• Income

• Number of items sold

• Temperature in Kelvin

Descriptive Statistics

summarize and describe the characteristics of a dataset, providing insights into its distribution, shape, and variability

Frequency Distribution

visualize the spread and patterns of scores

Simple FD

individual scores have been used and the data have not been grouped

Grouped FD

test-score intervals (class intervals) replaced the actual test scores

Frequency Distribution Tables

display the frequency of occurrence of different values or ranges within a dataset

Histograms

bar charts that show the frequency distribution of numerical data; useful for visualizing the shape, central tendency, and spread of the data

Bar Charts

used for categorical data to show the frequency or proportion of different categories

Pie Charts

used to represent parts of a whole for categorical data

Scatter Plots

used to visualize the relationship between two numerical variables

Line Graphs

ideal for displaying trends over time

Box Plots (Box-and-Whisker Plots)

display the median, quartiles, and potential outliers, providing a concise summary of the data’s distribution

Measures of Central Tendency

summary of measure that attempts to describe a whole set of data with a single value that represent the middle or center of its distribution

Mean

most stable and useful measure of central tendency

Median

the middle score in the distribution, ordinal, and useful for skewed distribution

Mode

• the most frequently occurring score in a distribution of scores

• Nominal data

Nominal

Measured Use: Mode

Ordinal

Measured Use: Median

Interval/Ratio (skewed)

Measured Use: Median

Interval/Ratio

Measured Use: Mean

Measures of Variability

summary measure that attempts to describe a whole set of data based how spread out the scores in a distribution

Range

equal to the difference between highest and lowest scores

Interquartile range

equal to the difference between Q3 and Q1

Semi-interquartile Range

equal to the interquartile divided by 2

Standard Deviation

equal to the square root of the average squared deviation about the mean

Variance

equal to the average of the square of the difference between the scores in the distribution and the mean

Measure of Location

summary of measure that attempts to describe a whole set of data on its location in the distribution

Percentile

used to display position or rank

Percentage

means of comparing quantities

Quartile

set of values which has three points dividing the data set into four identical parts

Decile

a quartile that groups the data into 10 equal parts

Skewness

• the degree of asymmetry observed in a probability distribution; imbalance in the distribution

• Univariate: -3 to +3

Positive Skew

relatively few of the scores fall at the high end of the distribution (difficult test)

Positive Skew

mean>median>mode

Negative Skew

relatively few of the scores fall at the low end of the distribution (easy test)

Negative Skew

mean<median<mode

Kurtosis

• a measure of the tailedness of a distribution; how often outliers occur

• Univariate: -10 to +10

Kurtosis

• The optimal boundary lines for the “upper” and “lower” areas of distribution scores will demarcate the upper and lower 27% of distribution scores

• 27% if normal

• 33% if platykurtic

Platykurtic

• relatively flat

• Thin-tailed, meaning that outliers are infrequent

Leptokurtic

• relatively peaked

• Fat-tailed, meaning that there are a lot of outliers

Mesokurtic

• somewhere in the middle

• Medium-tailed, so outliers are neither highly frequent, nor highly infrequent

Normal Curve

Bell-shaped, smooth, mathematically defined curve that is highest at its center, perfectly symmetrical yet asymptotic

Normal Curve

Also known as “Gaussian Curve”

Normal Curve

Mean = median = mode

Tail

the area on the normal curve between 2 and 3 standard deviation above and below the mean

Area Under the Normal Curve

50% of the score occur above the mean and 50% of the scores occur below the mean

Area Under the Normal Curve

Approximately 34% of all scores occur between the mean and 1 standard deviation above the mean

Area Under the Normal Curve

Approximately 34% of all scores occur between the mean and 1 standard deviation below the mean

Area Under the Normal Curve

Approximately 68% of all scores occur between the mean and

Area Under the Normal Curve

±1 standard deviation

Area Under the Normal Curve

Approximately 95% of all scores occur between the mean and ±2 standard deviations

Standard Scores

raw score that has been converted from one scale to another scale

Standard Scores

It tells us the position of test taker’s performance relative to other test taker’s readily apparent

Nonlinear Transformation

required when the data under consideration are not normally distributed yet needed to compare with normal distribution

Linear Transformation

retains a directed numerical relationship to the original raw score

Outlier

extremely atypical point located at a relatively long distance from the rest of the coordinated points in a scatterplot

Z-Score

Mean - 0

Standard Deviation - 1

T-Score

Mean - 50

Standard Deviation - 10

STANINE

Mean - 5

Standard Deviation - 2

STEN

Mean - 5.5

Standard Deviation - 2

IQ

Mean - 100

Standard Deviation - 15

IQ Subtest

Mean - 10

Standard Deviation - 3

GRE or SAT

Mean - 500

Standard Deviation - 100

Inferential Statistics

uses sample data to make conclusions or predictions about a larger population or dataset

Hypothesis Testing

an educate guess that sets the direction of the of the research; initial response

Null Hypothesis (H0)

assumes that there is no statistically significant relationship; default

Alternative Hypothesis (Ha)

assumes that there is a statistically significant relationship

Statistical Tests

is method of statistical inferences used to decide whether the data is sufficiently support a particular hypothesis; statistical treatment

Parametric Tests

class of statistical tests that make specific assumptions about the parameters of the population distribution from which the sample data are drawn

Nonparametric Tests

are statistical tests that do not rely on specific assumptions about the shape or parameters of the population distribution; “distribution-free”

Normal

Assumed Distribution - Parametric

Any

Assumed Distribution - Non-Parametric

Homogenous

Assumed Variance - Parametric

Heterogenous

Assumed Variance - Non-Parametric

Interval/Ratio

Typical Data - Parametric

Nominal/Ordinal

Typical Data - Non-Parametric

Independent

Data Set Relationship - Parametric

Any

Data Set Relationship - Non-Parametric

Pearson r Independent & Dependent T-Test Repeated Measures ANOVA One-/Two-way ANOVA

Statistical Tool - Parametric

Spearman rho Wilcoxon Signed Mann Whitney U Test Friedman Test Kruskal-Wallis Test

Statistical Tool - Non-Parametric

Measure of Bivariate Correlation

connection between 2 variables

Correlation

the degree and direction of correspondence between two things

Correlation

• Positive: ↑↑ or ↓↓

• Negative: ↑↓ or ↓↑

• No Correlation: —

Coefficient of Determination (r r )

analyzes how the differences in first variable can be explained by a difference in a second variable

Coefficient of Alienation

proportion of variance in the DV that is not accounted for by the IV