Measures of Central Tendency, Dispersion, and Location

• Measures of Central Tendency (Mean, Median, Mode)

• Measures of Dispersion (Range, IQR, Variance, Standard Deviation, CV)

• Measures of Location (Percentiles, Deciles, Quartiles)

What are the three main categories of summary measures in statistics?

measures of central tendency

help summarize large datasets by identifying a single representative value for a variable.

measures of dispersion

describe how spread out the data is around the central value.

Qualitative variables

Categories without numerical meaning (e.g., gender, eye color).

Quantitative variables

Numeric values that can be measured (e.g., height, weight).

Mean, Median, and Mode

What are the three measures of central tendency?

mean

average value of a dataset.

When there are extreme outliers or skewed distributions, as it is sensitive to extreme values.

When should the mean not be used?

median

middle value of an ordered dataset.

middle value

f n is odd: Median =

average of two middle values

If n is even: Median

mode

most frequently occurring value in a dataset.

Mean = Median = Mode

Normal Distribution:

Mean < Median < Mode

Left-Skewed Distribution:

Mode < Median < Mean

Right-Skewed Distribution:

measures of location

To determine the relative position of a value within a dataset.

percentiles, deciles, quartiles

What are the three main measures of location?

p-th percentile

the value below which p% of the observations fall.

IQR=Q3−Q1

How do quartiles relate to the Interquartile Range (IQR)?

measures of dispersion

They show how spread out the data is, indicating its variability.

range

difference between the highest and lowest values.

Variance

measures the average squared deviation from the mean.

standard deviation (SD)

square root of the variance, showing the average deviation from the mean.

s=s2s = \sqrt{s^2}

CV

compares the relative variability between datasets with different units.

Central tendency

gives the typical value, while dispersion tells us how much data varies around it.

Outliers

greatly affect the mean and variance but have little effect on the median and IQR.