Describing Variation & Distribution of Data

Variable

Variable

A measure of a single characteristic that can vary.

Causes of Variations

Factors such as biologic differences, genes, nutrition, environmental exposures, age, sex, race, presence or absence of disease, and extent of disease that contribute to variations in medical data.

Measurement Error

Errors in measurement techniques that can lead to variations in results.

Quantitative Data

Data represented by numbers and measurements.

Qualitative Data

Data represented by words.

Types of Variables

Nominal, Dichotomous, Ordinal, Continuous, and Ratio variables.

Frequency Distributions

Tables showing the frequency of values in a variable.

Range of a Variable

The difference between the lowest and highest observations of a variable.

Parameters of a Frequency Distribution

Measures of Central Tendency (Mean, Median, Mode) and Measures of Dispersion (Mean Absolute Deviation, Variance, Standard Deviation).

Skewness

Horizontal stretching of a frequency distribution leading to longer tails on one side (left or right).

Kurtosis

Vertical stretching or flattening of a frequency distribution.

Variable

A measure of a single characteristic that can vary

Causes of Variations

Biologic differences
Presence or absence of disease and extent of disease
Different conditions of measurement
Different techniques of measurement
Measurement error

Biologic differences

Genes, Nutrition, Environmental, Exposures, Age, Sex, Race

Different conditions of measurement

Often account for the variations observed in medical data

Measurement error

Can also cause variation

Types of Errors

Systematic Error and Random Error

Systematic Error

Can distort data systematically in one direction.

Random Error

Does not introduce bias

Quantitative Data 

Numbers and measurement

Qualitative Data

Generally use words

Nominal Variables

Naming or categoric variables that are not based on measurement scales or rank order.

Dichotomous (Binary) Variables

Variables with only two levels

Ordinal (Ranked) Variables

Data that can be characterized in terms of three or more qualitative values

Continous (Dimensional) Variables

Observation differs over time

Ratio Variables

If a continous scale has true 0 point

Frequency Distributions of Continuous Variable

Can be shown by creating a table that lists the values of the variable according to the frequency with which the value occurs.

Range of a variable

Range is the distance between the lowest and highest observations of the variable.

Real Frequency Distributions

Obtained from actual data or sample

Theoretical Frequency Distributions

Calculated using assumptions about the population from which the sample was obtained

Normal Distribution

Bell-shaped curve

Normal Distribution

Also called the Gaussian distribution after Johan Karl Gauss

Measures of Central Tendency

Mean
Median
Mode

Mean

Average value

Median

Middlemost or halfway value

Mode

Most frequent value

Mean Absolute Deviation

Does not have mathematical properties (as based form many statistical tests)

Variance

Standard Deviation

Square root of the variance

Standard Deviation

Used to describe the amount of spread in the frequency distribution

Standard Deviation

Average of deviations from the mean

Skewness

A horizontal stretching of a frequency distribution to one side or the other, so that one tail of observations is longer and has more observations than the other tail

Skewed to the left

When histogram or a frequency polygon has a longer tail on the left side of the diagram

Skewed to the left

Negatively skewed distribution

Skewed to the right

When histogram or a frequency polygon has a longer tail on the right side of the diagram

Skewed to the right

Positvely skewed distribution

Kurtosis

Characterized by a vertical stretching or flattening of the frequency distribution

Continous (Dimensional) Variables

Continous scales

Leptokurtic

Distribution with heavy tails.

Platykurtic

Distribution with light tails.

Mesokurtic

Distribution with moderate tails, similar to a normal distribution.

Graphs

It provide a visual way to understand the distribution and variation in the data.

Histogram

A bar graph that shows the frequency of data points within specified ranges (bins).

Box Plot (Box-and-Whisker Plot)

Displays the median, quartiles, and potential outliers. It helps visualize the spread and skewness of the data.

Dot Plot

Shows individual data points and their frequency.

Stem-and-Leaf Plot

Similar to a histogram but retains the original data values.

Density Plot

A smoothed version of the histogram, often used to estimate the probability density function of the data.

Five-Number Summary

Consists of the minimum, Q1, median, Q3, and maximum.

Summary Table

Includes mean, median, mode, range, variance, standard deviation, and other relevant statistics.

Outliers

Data points that significantly differ from the rest of the dataset.

Side-by-Side Box Plots

Useful for comparing the spread and central tendency of multiple groups.

Multiple Histograms

Placing histograms side by side or overlaying them for comparison.

Summary Statistics Comparison

Comparing means, medians, ranges, and standard deviations.