descriptive stat (Measures of Variability &Measures of Skewness and Kurtosis)

Measures of variability

(also measures of spread or dispersion) describe how similar or varied the set of observed values are for a particular variable (data item).

Range

is the difference between the lowest and highest observations.

Interquartile

is the difference between the lower quartile and the upper quartile.

Variance:standard deviation

_______ and ______ are essential statistical measures in biostatistics, providing insight into the spread or dispersion of continuous data such as blood pressure readings, patient recovery times, or cholesterol levels (Rosner, 2015).

Variance

Involves all observations in the distribution rather than through extreme observations

The average of the squared deviations from the mean.

Higher

A “_____ “variance means the data is more spread out.

lower

A ______ variance indicates that values are closer to the mean

Standard Deviation

Square root of the variance

• a measure of how much data values deviate away from the mean

Smaller

“"_____”" standard deviation (σ)→ Data is consistent and closely clustered around the mean.

Larger

“_____” standard deviation (σ)→ Data is more dispersed, indicating greater variability in values

Low (σ < 5% of mean)

Standard Deviation Size, Data points are closely packed; results are consistent

Moderate (σ ≈ 10% of mean)

Standard Deviation Size. Some variability, but most values stay close to the mean.

High (σ > 20% of mean)

Standard Deviation Size, Data is widely spread; there is significant variation.

Coefficient of variation

The ratio of the standard deviation to its mean

• A measure of relative dispersion – used to compare data sets with different units.

Standard Score

measures how many standard deviations a value is above or below the mean. • Alsoa measure of relative dispersion and position Formula

Coefficient of Variation (CV)2

*Purpose: Measures relative variability

When to Use: When comparing variability across different datasets or scales

Example: Comparing volatility of different stocks, assessing consistency in medical measurements

Standard Score (Z Score)2

*Purpose: Measures how far a value is from the mean

When to Use: When standardizing data or identifying outliers

Example: Checking if a student's test score is above or below average, detecting anomalies in data Measures of Skewness and Kurtosi

Skewness

Degree of asymmetry of a distribution

• Indicates not only the amount of asymmetry but also the direction of the distribution

• A distribution is said to be skewed in the direction of the extreme values, or speaking in terms of curve, in the direction of the excess tail

Positive (or right)

________ skewness refers to the distribution wherein the longer tail is directed to the right.

Negative (or left)

_______ skewness refers to the distribution wherein the longer tail is directed to the left.

zero skewness

means symmetrical distribution.

Kurtosis

A measure of the degree of peakedness or flatness relative to a normal distribution.

Leptokurtic

distribution having a relatively high peak (lepto– thin)

Mesokurtic

–normal distribution (meso–middle)

Platykurtic

distribution having a relatively flat top (platy – flat)

Skewness2

*When to Use: When checking for asymmetry in data distribution.

Example: Identifying biases in survey responses (e.g., Likert scale)

• Analyzing financial returns (e.g., stock market gains/losses)

• Checking normality assumptions before applying parametric tests (e.g., regression, ANOVA) Measures of Central Tendency Measures of Position

• Evaluating business sales data (e.g., right-skewed revenue growth

Kurtosis2

*When to Use: When assessing the tailedness of data and identifying extreme values (outliers).

example:

Detecting outliers in test scores or experiment results

• Evaluating risk in finance (e.g., high kurtosis indicates potential for extreme losses)

• Examining variability in manufacturing processes (e.g., product defect rates)

• Identifying inconsistencies in social science research (e.g., extreme Likert scale responses)

Skewness3

~It Measures Asymmetry of data distribution

Kurtosis3

~It measures. Tailedness of data distribution