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Educational Statistics Measures of Central Tendency and Variability (Dispersion) Over the Counter Drug Sales (in Millions) Tylenol 855 Mylanta 135 Advil 360 Tums 135 Vicks 350 Excedrin 130 One Touch 220 Benadryl 130 Robitussin 205 Halls 130 Bayer 170 Metamucil 125 Alka-Seltzer 160 Sudafed 115 Centrum 150 What is the AVERAGE number of drug sales? Measures of Central Tendency Measures of central tendency tell us something about the “typicalness” of a set of data. • Tell us what the typical score is in a distribution of scores. • Three measures of central tendency: – Mode – Median – Mean Measures of central tendency: the mode The mode is the score that occurs most frequently in a distribution. Sometimes more than one score occurs at frequencies distinctively higher than other scores, in which case there is(are) more than one mode: – Bi-modal distributions. – Multi-modal distributions. Only appropriate measure of central tendency for nominal data. Over the Counter Drug Sales (in Millions) Tylenol 855 Mylanta 135 Advil 360 Tums 135 Vicks 350 Excedrin 130 One Touch 220 Benadryl 130 Robitussin 205 Halls 130 Bayer 170 Metamucil 125 Alka-Seltzer 160 Sudafed 115 Centrum 150 What is the MODE of this distribution? Measures of central tendency: the median (Md) The median is the middle score in an ordered distribution of scores. It is the score that divides a distribution in half. It is also the score at the 50th percentile rank. The median can be found by computing the median location: (N + 1)/2. The median is the most appropriate measure of central tendency for ordinal data. Measures of central tendency: the median (Md) In the distribution given Score Freq Cum. F %c to the right, find the 6 2 2 3.8 median. 8 5 7 13.5 What is the percentile 9 0 7 13.5 rank (PR) of a score of 10 8 15 28.8 13? 11 11 26 50.0 12 9 35 67.3 What is the score cor- 13 6 41 78.8 responding to a 14 4 45 86.6 percentile rank of 29? 15 5 50 96.2 16 2 52 100.0 Over the Counter Drug Sales (in Millions) Tylenol 855 Mylanta 135 Advil 360 Tums 135 Vicks 350 Excedrin 130 One Touch 220 Benadryl 130 Robitussin 205 Halls 130 Bayer 170 Metamucil 125 Alka-Seltzer 160 Sudafed 115 Centrum 150 What is the Median of this distribution? Measures of central tendency: the Mean (µ,M, or ) • The most widely used measure of central tendency: • In words, the mean ( ) is the sum (Σ) of the scores (the X’s) divided by the number of scores (N). Over the Counter Drug Sales (in Millions) Tylenol 855 Mylanta 135 Advil 360 Tums 135 Vicks 350 Excedrin 130 One Touch 220 Benadryl 130 Robitussin 205 Halls 130 Bayer 170 Metamucil 125 Alka-Seltzer 160 Sudafed 115 Centrum 150 What is the Mean of this distribution? Measures of Central Tendency with special Distributions • The mode and bimodal distributions. – For distributions with more than one mode, the other measures of central tendency are misleading. • The Median and skewed distributions. – When a distribution is skewed the use of the mean may be misleading – Skew can be determined by the relative positions of the mean, median, and mode. Measures of Variability • How would you describe the 410 500 variability in the distribution of SAT-V 450 515 scores given at the right? • In other words, how 465 535 “spread-out” are the scores? 485 545 • Think about it. 500 585 • Write these values down. Measures of Variability • Three common measures of variability are – The Range. – The Variance. – The Standard deviation. • Other measures of variability are – The interquartile range. – The quartile deviation or semi-interquartile range. Measures of Variability The Range: (R) • R = The difference between the largest value in the distribution and the smallest value in the distribution. • I.e. R = Xlargest – Xsmallest. • Compute the Range for the distribution given. • R = 175. Measures of Variability The Variance (Var): • The variance is more computationally comples. • Defined as the average squared deviation from the mean of the distribution. • In symbols: Computing the Variance • First, compute X X the sum: 410 500 450 515 465 535 485 545 500 585 Computing the Variance • First, compute X X the sum: 410 500 450 515 • Then, divide by 465 535 N: 485 545 500 585 Computing the Variance • Next, subtract the X d X d mean from each score (call these 410 500 deviations from the mean, or, d ): 450 515 465 535 485 545 500 585 Computing the Variance • Next, subtract the X d X d mean from each score (call these 410 -89 500 1 deviations from the mean, or, d ): 450 -49 515 16 465 -34 535 36 485 -14 545 46 500 1 585 86 Computing the Variance • Next, Square the d d2 d d2 deviations from the -89 1 mean: -49 16 -34 36 -14 46 1 86 Computing the Variance • Next, Square the d d2 d d2 deviations from the -89 7921 1 1 mean: -49 2401 16 256 -34 1156 36 1296 -14 196 46 2116 1 1 86 7396 Computing the Variance Now, sum the d d2 d d2 squared deviations: -89 7921 1 1 -49 2401 16 256 -34 1156 36 1296 And divide by N: -14 196 46 2116 1 1 86 7396 Measures of Variability The Standard Deviation: Generally, we would prefer a measure of variability that tells us something about how far, on average, scores deviate from the mean. This is what the standard deviation tells us. Since the variance is the average squared deviation from the mean, the standard deviation, computed as the square root of the variance gives us the average deviation from the mean. Measures of Variability The Coefficient of Variation (CV) Distributions with larger means tend to have larger variances (and SDs) than distributions with smaller means. The CV provides convenient way to compare the variances of two or more distributions. SD CV X Using Statistics as Estimators We are rarely interested in sample statistics…we are interested in population parameters. Statistics are used to estimate (or make inferences about) parameters. The best statistics are sufficient, unbiased, efficient, and robust (or resistant) End of Presentation

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posted: | 8/17/2010 |

language: | English |

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