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