Chapter 3 Notes

©2008 by S. Gramlich

Updated 5/15/2014

Definitions and Formulas for Describing Data Numerically

! = Important Note

! These Notes are not meant to replace Reading.  Read Chapter first.

 

3-2

Mean  = average

            population: μ = ΣX / N

            sample: xbar = Σx / n

 

Median = middle #

Mode = most frequently occurring #

Midrange = (Max + Min) / 2

 

! Skip Mean from Frequency Distribution and Weighted Mean in Reading

 

Skewness (see Chapter 2 Notes)

 

3-3

Range = hi - lo

 

Variance & Standard Deviation = how far away, on average, the other data points are from the mean

 

Variance (in squared units)

! Recommend using (Definitional) Formula 3-4 from the text

            population σ2 = Σ(X-μ)2 / N

            sample s2 = Σ(x-xbar)2 / (n -1)

the numerator Σ(x-xbar)2 is called the Sum of Squared Deviations (SS)

the denominator (n-1) is called the degrees of freedom (df)

 

Standard Deviation (in singular units) is the square root of Variance

            population (Greek lowercase Sigma) σ = √σ2

            sample s = √s2

 

Empirical Rule

68% of data within xbar ± 1*s

95% of data within  xbar ± 2*s

99.7% of data within xbar ± 3*s

 

! Skip Range Rule of Thumb, Chebyshev's Theorem, and Coefficient of Variation in Reading

 

3-4

z score (standard score) = how many standard deviations away a score is from the mean

            population z = (X-μ) / σ

            sample z = (x-xbar) / s

! always round z score off to 2 decimal places

 

Boxplot = graph of 5-# summary

min = lo value

max = hi value

Quartile 1 = Q1 = P25

Quartile 2 = Q2 = P50 (median)

Quartile 3 = Q3 = P75

! to find Location of Quartile in data set, see flowchart (p. 116, fig 3-5) of text

 

!Skip Modified Boxplot and IQR in Reading

 

TECHNOLOGY

using StatCrunch: 

for 5-# summary, mean, variance, & standard deviation:  Stat - Summary Stats - Columns - Select Column - Calculate

for Boxplot:  Graphics - Boxplot - Select Column - Create Graph

 

EXCEL commands:

! highlight the data you want to calculate inside the parentheses

Statistic                                   Excel Command

Sum:                                        =sum( )

Sample Size:                            =count(  )

Minimum:                                =min( )

Maximum:                               =max( )

Mean:                                      =average( )

sample Variance:                     =var( )

sample Standard Deviation:    =stdev( )

Median            :                       =median(  )

Mode:                                      =mode( )

k-th Percentile:                        =percentile( ,.k)

k-th Quartile:                           =quartile(  ,k)