Measures of Central Tendency

Measures of Central Tendency

Mode

• Unimodal $\to$ data set has one mode

• Bimodal $\to$ data set has two modes

• Multimodal $\to$ many modes

• For the mean of a population we use $\mu$
• For the mean of a sample we use $\bar x$

• $\displaystyle\boxed{\overline{x}=\frac{\sum_{ }^{ }x_i}{n}}$ Sample mean
• $\displaystyle\boxed{\mu=\frac{\sum_{ }^{ }x_i}{N}}$ Population mean

• Example #1 (No extreme values)

• The mean here is $\displaystyle \frac{2+4+6+8+10}{5}=6$
• Example #2 (Extreme value)

• The maximum value "100" is an extreme value or outlier.
• The mean here is $\displaystyle\frac{2+4+6+8+100}{5}=24$

Median

• Example #1 (Odd number of data)

• The median is the number right in the middle with an equal number of data on each side.
• The median here is 6.
• Example #2 (Even number of data)

• The median is in between the two numbers located in the middle.
• The median here is the average of 6 and 8, which is 7.

• Example #3 (Extreme value)

• The highest value is now 100.
• The median here is still 6.

Summary

• The median is a resistant measure because it is not easily changed when there are extreme values.
• The mean is not a resistant measure because it is easily changed when there are extreme values.