Intra-Individual Variability Intra-individual variability is greater among older adults (Morse 1993) –May be an indicator of the functioning of the central nervous system (Dykiert 2012)(Dykiert 2012) –Neurodegenerative diseases increase variability Alzheimer’s Parkinson’s Disease following a traumatic brain injury Test with Reaction Time Test – on_version5.swfhttp:// on_version5.swf –Record a series of reaction times –Variability of reaction times is important

Example data from reaction time test

Reaction Time embedded Excel sheet

Variability A measure of how spread out the scores are in a distribution Usually accompanies a measure of central tendency as basic descriptive statistics for a set of scores As a descriptive measure variability measures the degree to which the scores are spread out or clustered together in a distribution. As an important component of most inferential statistics variability provides a measure of how accurately any individual score or sample represents the entire population

Population Variability Effects on sampling –When the population variability is small “homogeneous” all of the scores are clustered close together individual score or sample will necessarily provide a good representation of the entire set. –When population variability is large “heterogeneous” scores are widely spread one or two extreme scores can give a distorted picture of the general population

Figure 4-1 (p. 105) Population distributions of adult heights and adult weights.

Selective breeding for taste preference Selective breeding of dogs or cats for coat color, body size and behavior has occurred for centuries. Rats selectively bred for low versus high saccharin drinking. –Low-saccharin-consuming (LoS) –High-saccharin-consuming (HiS) Compare taste preference of male and female rats from HiS and LoS lines. –Values in the table are phenotype scores which is taste preference adjusted for baseline water and body weight.

Measuring Variability Variability can be measured with –range is the total distance covered by the distribution, from the highest score to the lowest score –variance average square distance from the mean –standard deviation measures the standard distance between a score and the mean In each case, variability is determined by measuring distance.

The Range The range is the total distance covered by the distribution The range tells us the number of measurement categories. –from the highest score (MAX) to the lowest score (MIN) For continuous variables Upper Real Limit for X max – LRL for X min Scores from 1 to 5 ; 5.5 – 0.5 = 5 Alternative definition of range: –When scores are whole numbers or discrete variables with numerical scores, X max – X min + 1 Scores from 0 to 4 4 – = 5

The Range Commonly the range can be defined as the difference between the largest score (max) and the smallest score (min). Used by SPSS computer program Scores from 1 to = 4 OK for discrete but not continuous variables Which calculation is used usually does not matter Range usually is not an accurate measurement of variability. Completely determined by two scores (max – min) An extreme large or small score will inflate the range Can be used as an adjunctive measure with variance

Interquartile and Semi-interquartile Range Interquartile range: Range covered by the middle 50% of the distribution = (Q3–Q1) Q3 is the 75th percentile, Q1 is the 25th percentile Semi-interquartile range: half the interquartile range = (Q3–Q1) / 2 less likely to be influenced by extreme scores

The Standard Deviation Standard deviation measures the standard (average) distance between a score and the mean. –All of the scores are distributed around the mean –there is a distance between each score and the mean called a deviation; X -  = deviation score –The average of all these deviation scores is standard deviation OOOPS Can not calculate the average deviation this way Sum of deviation from the mean is always zero Sum of deviations divided by number of scores 0/4 = ???? Need a work around see example 4.2

Example 4.2 table of data from page 109 Population of N = 5 scores. Calculate the mean Calculate deviation from the mean Calculate Squared Deviation Calculate “SS” Sum of Squared Deviations ∑ (X –  ) = 0 SS = ∑(X-  ) 2 = 40 ∑ X = 30  = (∑ X)/N  = 30/5 = 6

Figure 4.3 (page 111) from example data 4.2 A frequency distribution histogram for a population of N = 5 scores. The mean for this population is µ = 6. The smallest distance from the mean is 1 point, and the largest distance is 5 points. The standard distance (or standard deviation) should be between 1 and 5 points. See example 4.2 for calculation table Sum of Squares SS = ∑(X-  ) 2 = 40

Figure 4.2 (page 109) The calculations of variance and standard deviation

Calculation of Standard Deviation for a Population For example 4.2 Calculate mean  = (∑ X)/N  = 30/5 = 6 Calculate (SS) ∑(X-  ) 2 = 40 Compute variance which is mean square –calculate mean of the squared deviation (SS) divided by N – Variance  2 = SS/N = 40/5 = 8 Compute Standard Deviation –Standard deviation is the square root of variance –Standard deviation  = √8 = 2.83

Figure 4.3 (page 111) from example data 4.2 A frequency distribution histogram for a population of N = 5 scores. The mean for this population is µ = 6. The smallest distance from the mean is 1 point, and the largest distance is 5 points. The standard distance (or standard deviation) should be between 1 and 5 points. Sum of Squares SS = ∑(X-  ) 2 = 40 Variance  2 = SS/N = 40/5 = 8 Standard deviation  = √  2 = √8 = 2.83

Population Variance and Standard Deviation Formulas Definitional formulas –Sum of Squares SS = ∑(X-  ) 2 –Population Variance  2 = SS/N –Population Standard deviation  = √ SS/N Computational formulas Sum of Squares –Easier with a hand calculator –Using example 4.2 x x2x SS = /5 SS = SS = 40

Relationship with Other Statistical Measures Variance and standard deviation are mathematically related to the mean. They are computed from the squared deviation scores (squared distance of each score from the mean). Median and semi-interquartile range are both based on percentiles and therefore are used together. When the median is used to report central tendency, semi-interquartile range is often used to report variability. Range has no direct relationship to any other statistical measure.

Figure 4-4 (p. 114) The population of adult heights forms a normal distribution. If you select a sample from this population, you are most likely to obtain individuals who are near average in height. As a result, the scores in the sample will be less variable (spread out) than the scores in the population. FREQUENCYFREQUENCY 84 inches 48 inches

Calculation of Standard Deviation for a Sample 1. Compute the deviation (distance from the mean) for each score. 2. Square each deviation. 3.Compute the mean of the squared deviation a)called the variance or mean square b)for samples a)sum of the squared deviations (SS) b)dividing by n - 1, rather than N a)n - 1, is know as degrees of freedom (df) b)used so that the sample variance will provide an unbiased estimate of the population variance 4.take the square root of the variance to obtain the standard deviation

Example 4.5 table from page 116 Sample of n = 7 scores from a sample ∑ (X – M) = 0∑(X-M) 2 = 36 ∑ X = 35 M =(∑ X)/n M = 35/7 = 5

Figure 4-5 (p. 116) using Example data 4.5 (p. 116) The frequency distribution histogram for a sample of n = 7 scores. The sample mean is M = 5. The smallest distance from the mean is 1 point, and the largest distance from the mean is 4 points. The standard distance (standard deviation) should be between 1 and 4 points or about 2.5. Sum of Squares SS = ∑(X-M) 2 = 36 Sample Variance s 2 = SS/n-1 = 36/6 = 6 Sample Standard Deviation s = √ s 2 = √ 6 = 2.45 APA uses SD for standard deviation so it would be SD = √ s 2 = √ 6 = 2.45

Degrees of Freedom (df) Sample variability is biased, tends to under report population variability Using n – 1 corrects for that bias Sample variance is an unbiased estimate of population variance For a single sample of size n; degrees of freedom is n-1 Used as the denominator when calculating sample variance s 2 = SS/n-1 BTW: Degrees of freedom refers to number of scores that are free to vary –For a sample of n = 3 and a M = 5, where the first two scores are X = 4 and X = 5, the third score must be? –X = 6, it has to be, i.e. has no freedom to vary –So when n = 3 there are only 2 degrees of freedom