Probability & statistics - part 5

1.1.   Measures of Location

A fundamental task in many statistical analyses is to estimate a location parameter for the distribution, that is to find a typical or central value that best describes the data. There are five main location measures in statistics.

·         Mean

·         Median

·         Mode

·         Percentiles

·         Quartiles

If the measures are computedfor 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 toas the point estimator of thecorresponding population parameter.

Mean:

Perhaps the most important measure of location is the mean, or average value for a variable. The sample mean     is the point estimator of the population mean µ. Sample mean for n observations x₁, x₂,. . . , x_n can be calculated as,

Example:  Apartment Rents

Seventy efficiency apartmentswere randomly sampled ina small college town.  Themonthly rent prices forthese apartments are listed below in ascending order.

425	430	430	435	435	435	435	435	440	440
440	440	440	445	445	445	445	445	450	450
450	450	450	450	450	460	460	460	465	465
465	470	470	472	475	475	475	480	480	480
480	485	490	490	490	500	500	500	500	510
510	515	525	525	525	535	549	550	570	570
575	575	580	590	600	600	600	600	615	615

Median:

The median of a data set is the value in the middle when the data items are arranged in ascending order. Whenever a data set has extreme values, the median is the preferred measure of central location. The median is the measure of location most often reported for annual income and property value data.   A few extremely large incomes or property valuescan inflate the mean.

For an odd number of observations, the median is the middle value when arranged in ascending or descending order and if even number of observations the median is the average of two middle values.

Example:  Apartment Rents

Averaging the 35^th and 36^th data values:

Median = (475 + 475)/2 =    475

Mode :

The mode of a data set is the value that occurs with greatest frequency. The greatest frequency can occur at two or more different values. If the data have exactly two modes, the data arebimodal. If the data have more than two modes, the data aremultimodal.

Example : Apartment Rents

450 occurred most frequently (7 times)

Mode =   450

Percentiles:

A percentile provides information about how thedata are spread over the interval from the smallestvalue to the largest value.Admission test scores for colleges and universitiesare frequently reported in terms of percentiles.The p^th 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.

Compute index i, the position of the p^th percentile.i = (p/100)*n. If i is not an integer, round up.  The p^th percentile is the value in the i^th position. If i is an integer, the p^th percentile is the average of the values in positions i and i +1.

Example : Apartment Rents

90^th Percentile :

i = (p/100)n = (90/100)70 = 63

Averaging the 63rd and 64th data values:

90th Percentile = (580 + 590)/2 =   585

At least 90%of the itemstake on a valueof 585 or less.”

Quartiles:

Quartiles are specific percentiles. First quartile is the 25^thpercentile, second quartile is the 50^th percentile (or the median) and third quartile is the 75^thPercentile.

Example : Apartment Rents

Third quartile = 75th percentile

i= (p/100)n = (75/100)70 = 52.5 = 53

Third quartile =   525

1.2.   Measures of Variability

In a data set, it is often desirable to consider measures of variability (dispersion), as well as measures of location. For example, suppose that you are a purchasing agent for a large manufacturing firm that you regularly place orders with two different suppliers. In choosing between the two suppliers we might consider not only the average delivery time for each, but also the variability in delivery time for each.

There are several measures of variability we can calculate for a data set to see the dispersion.

·         Range

·         Interquartile Range

·         Variance

·         Standard Deviation

·         Coefficient of Variation

Range:

The range of a data set is the difference between the largest and smallest data values. It is the simplest measure of variability and is very sensitive to the smallest and largest data values.

Example : Apartment Rents

Range = largest value - smallest value

Range = 615 - 425 =   190

Interquartile Range

The interquartile range of a data set is the difference between the third quartile and the first quartile which represents the range for the middle 50% of the data. It overcomes the sensitivity to extreme data values.

Example : Apartment Rents

3rd Quartile (Q3) = 525

1st Quartile (Q1) = 445

Interquartile Range = Q3 - Q1 = 525 - 445 =   80

Variance:

The variance is a measure of variability that utilizes all the data.  It is based on the difference between the value of each observation (x_i) and the mean (    for a sample, µ for a population).

The sample variance denoted by s², is the average of the squared differences between each data value and the mean and can be computed as follows:

Example : Apartment Rents

Standard Deviation:

The standard deviation of a data set is the positive square root of the variance. It is measured in the same units as the data, making it more easily interpreted than the variance.

The sample standard deviation is computed as,

Example : Apartment Rents

               

Coefficient of Variation:

The coefficient of variation indicates how large the standard deviation is in relation to the mean. The sample coefficient of variation is computed as,

Example : Apartment Rents