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Numerical Measures of Central Tendency Mean

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

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Numerical Measures of Central Tendency Mean

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  1. Numerical Measures of Central TendencyMean

  2. Symbols • Notation • Series of observations • Χ1 , Χ2 , Χ3 , Χ4 ,... Χn • Then Χ1 = 5, Χ2 = 7, Χ3 = 3, Χ4 = 8, Χ5 = 7

  3. Symbols • Notation • Sum of data values • Χ1 + Χ2 + Χ3 + Χ4 ... Χn • Σ symbol • Sum • Σx

  4. 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

  5. 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 =

  6. 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

  7. 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

  8. 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

  9. 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

  10. Mean • Using an equation…the mean is Sample Mean Population Mean This could be written as:

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

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

  13. 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 • Xmis equal to the midpoint of the class

  14. Grouped Data Set Midpoint Xm = (min + max)/2

  15. Grouped Data Set • = Σf(Xm)/n = 56/8 = 7 • = 7

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

  17. 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...)

  18. Weighted Mean

  19. Weighted Mean • = ΣwX/Σw = 32/12 • = 2.67

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

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