Numerical Measures of Central Tendency Mean

Symbols Notation – Series of observations Χ 1, Χ 2, Χ 3, Χ 4,... Χ n Then Χ 1 = 5, Χ 2 = 7, Χ 3 = 3, Χ 4 = 8, Χ 5 = 7 Observation12345 Data Value (Hours) 57387

Symbols Notation – Sum of data values Χ 1 + Χ 2 + Χ 3 + Χ 4... Χ n Σ symbol – Sum – Σx

Sum Σx = Σx = 30 Variations on the Sum – ΣX - Add all values – ΣX 2 – First, square all values, then sum – (ΣX) 2 - First, sum all values, then square the sum Observation12345 Data Value (Hours) 57387

Summation Examples Data set is: 5, 7, 3, 8, 7 ΣX = = ΣX 2 = = (ΣX) 2 = ( ) 2 =

Summation Examples Data set is: 5, 7, 3, 8, 7 ΣX = = 30 ΣX 2 = = 196 (ΣX) 2 = ( ) 2 = 900

Central Tendency Measures of central tendency are used to display the idea of centralness for a data set. Most common measures Mean Median Mode Midpoint Midrange

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 This could be written as: Sample MeanPopulation Mean

Mean Examples Data set is: 5, 7, 3, 8, 7 What is the ? Σx = = 30 n = 5 = Σx/n = 30/5 = 6 = 6

Mean Examples Data set is: 5, 7, 3, 8, 7, 15 What is the ? ΣX = = 45 n = 6 = ΣX/n = 45/6 = 7.5 = 7.5

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 X m is equal to the midpoint of the class

Grouped Data Set Class (lbs)fXmXm f(X m ) 0 – 4424(2) = 8 5 – 9277(2) = – (12) = – (17) = 0 20 – (22) = 22 Midpoint X m = (min + max)/2

Grouped Data Set Class (lbs)fXmXm f(X m ) 0 – 4424(2) = 8 5 – 9277(2) = – (12) = – (17) = 0 20 – (22) = 22 n = = 8Σf(X m )= = 56 = Σf(X m )/n = 56/8 = 7 = 7

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...)

Weighted Mean Coursew (Credit Hours) Grade (x)w(x) PSY 1013A – 4pts3(4)=12 BIO 1043C – 2pts3(2) = 6 BER 3454B – 3pts4(3)= 12 SPE 2402D – 1pt2(1) = 2

Weighted Mean Coursew (Credit Hours)Grade (x)w(x) PSY 1013A – 4pts3(4)=12 BIO 1043C – 2pts3(2) = 6 BER 3454B – 3pts4(3)= 12 SPE 2402D – 1pt2(1) = 2 Σw = Σw =12 ΣwX = ΣwX=32 = ΣwX/Σw = 32/12 = 2.67

Homework eLearning Assessments Central Tendency Homework 1 Due Next Class Meeting (accepted through eLearning until 10:00 am day of next class).