Objectives

To learn about:

• Measures of Center

• Measures of Variation

• Measures of Relative Standing

Some Related Definitions

• Center

• Variation

• Distribution

• Outliers

• Time

Measures of Center

• Mean and Weighted Mean

• Mode

• Median

• Range

• Midrange

Mean and Weighted Mean

Mean: Average

Definition:

Sample Mean:

Population Mean:

Weighted Mean Mean adjusted by weights of each data point

Definition:

Data multiplied by the corresponding weights

Divide by the sum of the weights

Weighted Mean Example

• Example: Grade calculation for the class

Mode • The data point or points that occur most

frequently in the dataset

• Can have more than one more; e.g. bimodal data set or multimodal data set

• Can have no mode at all

Mode Example

Median

• Middle of the data set when ordered

• If odd data points in the data set then the median is a singular data point that is in the middle

• If even number of data points then median is the average of the middle two data points

Median Example

• Typically used for data set that may contain extreme values such as home prices, yearly incomes etc.

Range and Midrange

• Highest data point minus the lowest data point

• Highest plus lowest divided by two

Standard Deviation

• Variation of the data points from the mean

• Measures how spread out the data points are relative to the mean value of the data set

• Measured by the following formula:

• Range rule of thumb for s

Examples of Standard Deviation

• Calculate standard deviation for these data points: {–2, 5, –8, 7}

• Normal Distribution example:

Standard Deviation Examples

• Sample standard deviation: s

• In the formula divide by (n – 1)

• Population standard deviation:

• In the formula divide by N

Variance

• Square of standard deviation

• Average of the square of the differences between mean and the data points

• Distinguish between sample variance and population variance:

2

Sample Variance

Population Variance

2s

Empirical Rule

• One standard deviation in either direction from mean

• Two standard deviations in either direction from mean

• Three standard deviations in either direction from mean

Examples of Empirical Rule

• Salinas yearly income bell shaped/normal with mean $48,000 and standard deviation $9000

• Find more than $57,000

• Less than $30,000

• Between $39,000 and $57,000

• Between $30,000 and $66,000

• Between $30,000 and $57,000

• More than $57,000 or less than $30,000

Chebyshev’s Rule

• If K is greater than 1 then at least

• Arbitrary distribution

• Does not work if K is less than or equal to 1

Chebyshev’s Rule Examples

• Salinas yearly income has mean $48,000 and standard deviation $9000

• Between $30,000 and $66,000

• Find more than $66,000

• Less than $30,000

• More than $61,500

• More than $66,000 or less than $30,000

Percentile

• Order data points in increasing order

• Find the corresponding data point by using the formula

• Always round up

• Example:

Boxplot

• Lower quartile

• Upper quartile

• Maximum and minimum