Numerical Measures - Statistics - Lecture Slides, Slides of Statistics

Download Numerical Measures - Statistics - Lecture Slides and more Slides Statistics in PDF only on Docsity!

Descriptive Statistics: Numerical

Measures

 Measures of Location (central tendency)

 Measures of Variability

Measures of Location

If the measures are computed for data from a sample, they are called sample statistics.

If the measures are computed for data from a population, they are called population parameters.

A sample statistic is referred to as the point estimator of the corresponding population parameter.

 Mean

 Median

 Mode

 Percentiles

 Quartiles

Sample Mean (^) x

Number of observations in the sample

Sum of the values of the n observations

x i

x

n

Population Mean μ

Number of observations in the population

Sum of the values of the N observations

x i

N

Sample Mean

34, 356 (^) 490. 70

x x^ i n

= ∑ = =

 Example: Apartment Rents

Median

 Whenever a data set has extreme values, the median is the preferred measure of central location.

 A few extremely large incomes or property values can inflate the mean.  Applicable for ordinal, interval, and ratio data  Unaffected by extremely large and extremely small values

 The median is the measure of location most often reported for annual income and property value data.

 The median of a data set is the value in the middle when the data items are arranged in ascending order.

Median

 For an even number of observations:

in ascending order

(^26 18 27 12 14 27 30) 8 observations

the median is the average of the middle two values.

Median = (19 + 26)/2 = 22.

Median

Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475

Note: Data is in ascending order.

 Example: Apartment Rents

Mode

450 occurred most frequently (7 times) Mode = 450

Note: Data is in ascending order.

 Example: Apartment Rents

Percentiles

 The pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more.  Not applicable for nominal data  Example: 90th percentile indicates that at least 90% of the data lie below it, and at most 10% of the data lie above it

 A percentile provides information about how the data are spread over the interval from the smallest value to the largest value.

 Admission test scores for colleges and universities are frequently reported in terms of percentiles.

80 th^ Percentile

i = ( p /100) n = (80/100)70 = 56 Averaging the 56 th^ and 57th^ data values: 80th Percentile = (535 + 549)/2 = 542

Note: Data is in ascending order.

 Example: Apartment Rents

80 th^ Percentile

“At least 80% of the items take on a value of 542 or less.”

“At least 20% of the items take on a value of 542 or more.” 56/70 = .8 or 80% 14/70 = .2 or 20%

 Example: Apartment Rents

Third Quartile

Third quartile = 75th percentile i = ( p /100) n = (75/100)70 = 52.5 = 53 Third quartile = 525

Note: Data is in ascending order.

 Example: Apartment Rents

Measures of Variability

 It is often desirable to consider measures of variability (dispersion), as well as measures of location.

 For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each.