Variable

•  A measure of a single characteristic that can vary

Causes of Variations:

• Biologic Differences

  • Genes

  • Nutrition

  • Environmental

  • Exposures

  • Age

  • Sex

  • Race

• Presence or absence of disease and extent of disease

  • Ex. Cancer of the cervix may be in situ, localized, invasive, or metastatic

• Different conditions of measurement

  • Often account for the variations observed in medical data

  • Factors: time of the day, ambient temperature or noise, and the presence of fatigue or anxiety in the patient

• Different techniques of measurement

  • Can produce different results

• Measurement error

  • Can also cause variation

Types of Errors:

• Systematic Error

  • Can distort data systematically in one direction.

  • Can introduce bias

• Random Error

  • Does not introduce bias

Quantitative Data 

• Numbers and measurement

Qualitative Data

• Generally use words

Types of Variables:

• Nominal Variables

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

  • Ex. Blood groups, occupations, skin color

• Dichotomous (Binary) Variables

  • Variables with only two levels

  • Ex. Study of heart murmurs (systolic or diastolic)

• Ordinal (Ranked) Variables

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

  • Ex. Satisfaction of care

• Continous (Dimensional) Variables

  • Continous scales

  • Observation differs over time

  • Ex. Height, Weight, Blood pressure

• Ratio Variables

  • If a continous scale has true 0 point

  • Ex. Kelvin Temperature

Frequency Distributions

• 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 and Theoretical Frequency Distributions

  • 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

    • Also called the Gaussian distribution after Johan Karl Gauss

    • Bell-shaped curve

• Parameters of a Frequency Distribution

  • Measures of Central Tendency 

    • Mean (x̄) – Average value

    • Median – Middlemost or halfway value

    • Mode – Most frequent value

  • Measures of Dispersion

    • Based on Percentiles

      • Percentile of Distribution

        • A point which at which a certain percentage of the observations lie below the indicated point when all the observations are ranked in descending order.

      

    • Based on Mean

      • Mean Absolute Deviation

        • Seldom used, but helps define the concept of dispersion

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

        

      • Variance

        • Fundamental measure of dispersion

        

      • Standard Deviation

        • Square root of the variance

        • Used to describe the amount of spread in the frequency distribution

        • Average of deviations from the mean

Problems in Analyzing a Frequency 

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

  • Negatively skewed distribution

• Skewed to the right

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

  • Positvely skewed distribution

• Kurtosis

  • Characterized by a vertical stretching or flattening of the frequency distribution

    

    • Leptokurtic: Distribution with heavy tails.

    • Platykurtic: Distribution with light tails.

    • Mesokurtic: Distribution with moderate tails, similar to a normal distribution.

Graphical Representations

Graphs 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.

Descriptive Statistics Summary

Combining various descriptive statistics provides a comprehensive overview 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

Outliers are data points that significantly differ from the rest of the dataset.

• Detection: Using methods such as the IQR (1.5*IQR rule) or Z-scores.

• Impact: Outliers can skew the results and give a misleading picture of the data distribution.

Comparing Distributions

Comparing different datasets involves looking at their central tendency, spread, and shape.

• 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.