Basic Concepts of Measure of Center

Mean of a set of data is the measure of center found by adding all of the data values and dividing the total by the number of data values. (not resistant)

Formula: (sample size = n)

$μ = \frac{\sum x}{n}$ for parameter

$M = \frac{\sum x}{n}$ for statistic

Median of data set is the measure of center that is the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude. (resistant)

Mode of a data set is the value that occur with the greatest frequency. (can be used for qualitative data)

Types of modes

1. unimodal

2. bimodal

3. multimodal

4. no mode(uniform or rectangular)

Midrange of a data set is measure of center that is the value midway between the maximum and minimum values of the original data set. (not resistant)

Formula:

midrange = $\frac{max + min}{2}$

Range of a set of data values is the difference between the maximum data value and the minimum data value.

Formula:

range = Max - min

Basic Concepts of Variation

Variance of a set of values is a measure of variation equal to the square of the standard deviation.

Formula:

$σ^{2} = \frac{\sum {(x - μ)}^{2}}{n}$ for parameter

$s^{2} = \frac{\sum {(x - M)}^{2}}{n - 1}$ for statistic

Standard deviation of a set of sample values, is a measure of how much data values deviate away from the mean.

Formula:

$σ = \sqrt{σ^{2}}$ for parameter

$s = \sqrt{s^{2}}$ for statistic

z-Scores, Percentiles, Quartiles and Boxplots

z-Score or standardized value is the number of standard deviations that a give value x (score) is above or below the mean.

Formula:

$z = \frac{x - μ}{σ}$ for parameter

$z = \frac{x - M}{s}$ for statistic

Percentiles are measures of location, denoted P₁, P₂, ...,P₉₉ which divide a set of data into 100 groups with about 1% of the values in each group.

formula to convert number x to percentile:

Percentile of value x = $\frac{n u m b e r o f v a l u e s l e s s t h a n x}{t o t a l n u m b e r o f v a l u e s} \times 100$

Convert Percentile to number:

L = $(\frac{k}{100}) n$

if L is a whole number then the value of the kth percentile is midway between the Lth value and the next value in the sorted list.
if L is not a whole number then round it up to next larger whole number and count from the from the lowest.

Quartiles are measures of location, denoted Q₁, Q₂ and Q₃, which divide a set of data into four groups with about 25% of the values in each group.( $P_{25}, P_{50}, P_{75}$ )

5- number summary consists of these five values:

1. minimum

2.First Quartile

3.Second Quartile (median)

4.Third Quartile

5. Maximum

Boxplot is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, the median and third quartile.

Ex:

Vertical Box Plot boxplot 2

Prarire Dog hill box plot