Numerical Measures of Central Tendency

Mean

Symbols

• Notation– Series of observations• Χ1 , Χ2 , Χ3 , Χ4 ,... Χn

• Then Χ1 = 5, Χ2 = 7, Χ3 = 3, Χ4 = 8, Χ5 = 7

Observation 1 2 3 4 5

Data Value (Hours)

5 7 3 8 7

Symbols• Notation– Sum of data values• Χ1 + Χ2 + Χ3 + Χ4 ... Χn

• Σ symbol– Sum– Σx

Sum

• Σx = 5 + 7 + 3 + 8 + 7• Σx = 30• Variations on the Sum– ΣX - Add all values– ΣX2 – First, square all values, then sum– (ΣX)2 - First, sum all values, then square the sum

Observation 1 2 3 4 5Data Value (Hours)

5 7 3 8 7

Summation Examples

• Data set is: 5, 7, 3, 8, 7

• ΣX = 5 + 7 + 3 + 8 + 7 =

• ΣX2 = 52 + 72 + 32 + 82 + 72=

• (ΣX)2 = (5 + 7 + 3 + 8 + 7) 2 =

Summation Examples

• Data set is: 5, 7, 3, 8, 7

• ΣX = 5 + 7 + 3 + 8 + 7 = 30

• ΣX2 = 52 + 72 + 32 + 82 + 72 = 196

• (ΣX)2 = (5 + 7 + 3 + 8 + 7) 2 = 900

Central Tendency

Measures of central tendency are used to display the idea of centralness for a data set.

Most common measuresMeanMedianModeMidpointMidrange

Mean

The mean is the arithmetic average of the values in a distribution.

• Uses all the data values

• Influenced by extreme values (high/low) called outliers

• Used to calculate other statistics

• Value is unique and may not be a data value

The Mean

• It is sometime called the arithmetic mean

• This is computed by summing up all of the scores and dividing by the total number of observations

• Using an equation…the mean is Σx/n– Where n is equal to the total number of

observations in your data set

Mean

• Using an equation…the mean is

1 2 3 1n nx x x x xx

n

This could be written as:

n

XX

n

X

Sample Mean Population Mean

Mean Examples

• Data set is: 5, 7, 3, 8, 7What is the ?

• Σx = 5 + 7 + 3 + 8 + 7 = 30• n = 5• = Σx/n = 30/5 = 6• = 6X

X

X

Mean Examples

• Data set is: 5, 7, 3, 8, 7, 15What is the ?

• ΣX = 5 + 7 + 3 + 8 + 7 + 15= 45• n = 6• = ΣX/n = 45/6 = 7.5• = 7.5

X

X

X

Mean for Grouped Data

• When our data is grouped or is formatted in a frequency table, we can use a separate formula for calculating the mean:

• f is equal to the frequency of the class• Xm is equal to the midpoint of the class

n

XfX m )(

Grouped Data Set

Class (lbs) f Xm f(Xm)

0 – 4 4 2 4(2) = 85 – 9 2 7 7(2) = 1410 – 14 1 12 1(12) = 1215 – 19 0 17 0(17) = 020 – 24 1 22 1(22) = 22

Midpoint Xm = (min + max)/2

Grouped Data SetClass (lbs) f Xm f(Xm)

0 – 4 4 2 4(2) = 8

5 – 9 2 7 7(2) = 14

10 – 14 1 12 1(12) = 12

15 – 19 0 17 0(17) = 0

20 – 24 1 22 1(22) = 22

n = 4+2+1+1 = 8 Σf(Xm )= 8+14+12+0+22 = 56

• = Σf(Xm)/n = 56/8 = 7• = 7X

X

Weighted Mean

• When the values are not represented equally then the use a weighted mean is required• GPA• Weighted by the credit hours

Weighted Average

• We include the weightings into our calculation of the mean

• w = weight (ex. Credit hours)• x = grade (for each course A = 4, B = 3, etc...)

w

wXX

Weighted Mean

Course w (Credit Hours)

Grade (x) w(x)

PSY 101 3 A – 4pts 3(4)=12BIO 104 3 C – 2pts 3(2) = 6BER 345 4 B – 3pts 4(3)= 12SPE 240 2 D – 1pt 2(1) = 2

w

wXX

Weighted MeanCourse w (Credit Hours) Grade (x) w(x)PSY 101 3 A – 4pts 3(4)=12BIO 104 3 C – 2pts 3(2) = 6BER 345 4 B – 3pts 4(3)= 12SPE 240 2 D – 1pt 2(1) = 2

Σw = 3+3+4+2 Σw =12

ΣwX = 12 + 6 + 12 + 2 ΣwX=32

w

wXX

• = ΣwX/Σw = 32/12 • = 2.67X

X

HomeworkeLearning AssessmentsCentral Tendency Homework 1Due Next Class Meeting (accepted through

eLearning until 10:00 am day of next class).