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Measures of Central Tendency Measures of Central Tendency X Mean (Arithmetic Average) Mean Computed from raw scores. N X X 1 N Mean (Arithmetic Average): Raw Score Formula Score Coach John Buchanan 55 Holt Cornell 46 Robert Turner 50 Charlie King 51 Jim McCright 48 John Coate 50 Ed Passmore 30 David Moser 53 Jack Patton 57 Jane Benedict 62 Coach William Foster David O'Steen Bill Collmer John Kopplin Tom Tuley Mike Bratcher John Achor Joseph Karels Lewis Flynn Brent Stephens N= Sum= Score 39 47 52 54 48 46 68 44 49 52 30 1493 Coach Frank Young Dave Abbott Danny Cooper Ron Clarke Charles Gerbing Winston Edwall Dean Wyman Rinehart Slife Roger Frish Michael Barker Mean= Score 45 33 50 51 54 59 49 42 56 53 49.77 Measures of Central Tendency Mean (Arithmetic Average) X Mean Computed from grouped scores. (A frequency Distribution) k X fX 1 N mp = 3408.5 = 33 103.29 IQ ƒ MP ƒMP 150-159 140-149 130-139 120-129 110-119 100-109 90-99 80-89 70-79 60-69 50-59 1 0 2 3 6 8 5 5 1 1 1 154.5 144.5 134.5 124.5 114.5 104.5 94.5 84.5 74.5 64.5 54.5 154.5 0.0 269.0 373.5 687.0 836.0 472.5 422.5 74.5 64.5 54.5 N= 33 Sum= 3408.5 Mean= 103.29 Measures of Central Tendency Mean (Arithmetic Average) X Mean Computed from Guessed Average k X X GA i ( fx ' ) 1 N 4 104.5 (10) 104.5 (10)(.121) 10329 . 33 IQ ƒ x' ƒx' 150-159 140-149 130-139 120-129 110-119 100-109 90-99 80-89 70-79 60-69 50-59 1 0 2 3 6 8 5 5 1 1 1 5 4 3 2 1 0 -1 -2 -3 -4 -5 5 0 6 6 6 0 -5 -10 -3 -4 -5 33 Sum= -4 N= Measures of Central Tendency The Mean (Average) • Advantages – Most stable measure – Can perform algebraic operations – Basis for advanced statistics – Gives us the Center of Gravity of a dist. – Value depends on every score in dist. • Disadvantages – Weights extreme scores more than other measures of Central Tendency. To Summarize: • Calculations for both grouped and ungrouped data. • Use raw scores when possible • Use grouped formula to calculate from freq..... dist. or graph. Measures of Central Tendency • Guessed Average & Arbitrary Origin methods are seldom used today, but we need to be aware of them. •Raw score methods have the advantage of being more precise, because they use the exact value of every score in the distribution. • Next we will discuss - The Median The Median Measures of Central Tendency The Median Mdn The median is the point on the scale of measurement above which, and below which 50% of the scores are located. Measures of Central Tendency The Median Mdn 18 18 19 20 23 For raw scores the median is just the score in the middle. If no score fall in the middle, we just interpolate. Measures of Central Tendency The Median Mdn • Median Computed from Group Data n Mdn X l i 2 fi f b IQ 150-159 140-149 130-139 120-129 110-119 100-109 90-99 80-89 70-79 60-69 50-59 f Cum U 1 0 2 3 6 8 5 5 1 1 1 21 13 8 3 2 1 Cum D 1 1 3 6 12 20 Measures of Central Tendency The Median Mdn • Median Computed from Group Data n Mdn X l i 2 fi f b 33 13 99.5 10( 2 ) 10388 . 8 IQ 150-159 140-149 130-139 120-129 110-119 100-109 90-99 80-89 70-79 60-69 50-59 f Cum U 1 0 2 3 6 8 5 5 1 1 1 21 13 8 3 2 1 Cum D 1 1 3 6 12 20 Measures of Central Tendency The Median ... • Advantages – Easy to calculate. – Not influenced by extreme scores, so it can be used when we have extreme values. – The median divides the distribution into two equal groups. • Disadvantages – It is less stable than the mean. – The median will not permit all algebraic operations (addition & subtraction) because we usually have ordinal scales The Mode Measures of Central Tendency • The Mode – Ungrouped Data • The mode is the score which occurs most frequently. – Grouped Data • The mode is the midpoint of the class interval containing the largest number of cases. – Estimating the Mode from the Mean and Median • Mo = 3(Mdn)-2(Mean) • For use in skewed distributions. • Also distributions which are bimodal. Measures of Central Tendency The Mode ... • Advantages – Easy to calculate. – It is most appropriate for discreet Data. – It gives the most typical case. • Disadvantages – It is the least stable measure of central tendency. – Different size class intervals yield different results – If two non-adjacent class intervals have the same frequency, the distribution is bimodal and the mode is meaningless. • May suggest two distributions. – Can’t perform arithmetic or algebraic expressions with Mo. Measures of Central Tendency Next: Measures of Variability

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