Chapter 4: Variability
variability: provides a quantitative measure of the differences between scores in a distribution and describes the degree to which the scores are spread out or clustered together. particularly important for inferential statistics.
can also be seen as measuring predictability (you can predict relatively accurately where a score will fall), consistency (scores are relatively equal between studies/trials), or even diversity (differences between individuals within the study) w/in a data set.
2 key purposes of measuring variability:
describes the distribution of scores—are they clustered together or spread out? variability is usually defined in terms of distance—how much distance you can expect between one score and another, or how much difference to expect between an individual score and the mean.
measures how well an individual score (or group of scores) represents the entire distribution; ie, provides info about how much error to expect if using a sample to represent a population.
3 main measures of variability: the range, standard deviation, and the variance.
range: the distance covered by the scores in a distribution, from the largest to the smallest.
for data sets with well-defined upper and lower limits, it’s calculated by subtracting minimum from maximum.
for continuous variables, we define the range as the difference between the upper real limit (URL) for the largest score (Xmax) and the lower real limit (LRL) for the smallest score (Xmin): . eg, values from 1 to 5 cover 0.5 to 5.5 and thus cover a range of 5 points.
when scores are whole numbers, we can also define the range as the number of measurement categories. if people are either assigned a 1, 2, or 3, there are 3 measurement categories and the range is 3 points. this also works for discrete variables that are measured numerically. in this case, we can calculate the range with .
the problem with the range is that it ignores all scores between the max and min—if you have major outliers, the range doesn’t take that into account, and you don’t know that from just a range—there’s no accounting for whether there’s a cluster somewhere within that range. due to this, we rarely use range.
standard deviation is most common measure; uses the mean as a reference point and measures variability by considering distance between each score and the mean. provides a measure of the standard/average distance from the mean and describes whether scores are clustered together or not.
1st step to finding standard deviation: find the deviation or deviation score, the difference between a score and the mean, which is calculated with the equation. can be positive or negative!
2nd step: calculate the mean of the deviation scores. the scores for the complete set always add up to zero (because of the definition of mean), so this is really just a way to make sure you didn’t fuck something up yet.
3rd step: square all the scores from step 1; then, compute the average of the squared deviations, or the mean squared deviation, which is called variance (the mean of the squared deviations; the average squared distance from the mean).
note that due to the squaring, variance is helpful for inferential statistics but not for descriptive measures.
4th step: take the square root of the variance (). this is the standard deviation (the square root of the variance; provides a measure of the standard, or average distance from the mean).
note that you can estimate standard deviation based on the mean—if you look at a histogram, eg, and see that from the mean, the shortest distance of a score is 1 and the largest is 5, you can predict that it’s between 1 and 5, probably around 3. if your score falls within these parameters, you know you probably did the math right.
note that variance and standard deviation require numerical scores, so they’re only ever used with interval and ratio scales.
the calculations for variance and variance standard deviation are slightly different for populations versus samples despite the concepts being identical:
populations:
SS/sum of squares is the sum of the squared deviation scores [that is important to variability]. (recall that variance = mean squared deviation = .)
definitional formula: SS=Σ(X-μ)2. find each deviation score, square each deviation score, and add them all up. most direct formula, but can be difficult to use, esp if mean is fractional/decimal (esp because of rounding errors).
for cases with non-whole numbers, we use the computational formula, which uses the scores rather than the deviations and thus minimizes decimal/fraction complications: . square all the scores and add them up; then add up all the scores and square that separately, before dividing by N. finally, take your initial value and subtract that second value. it’s more complex but also more accurate.
note: these equations are equivalent! you can use either; it’s just that the latter is preferable over the former in the case of decimals. both should get you the exact same thing, assuming you did everything right!
we can now express population variance as and population standard deviation as .
population variance (σ2): the mean squared distance from the mean; obtained by dividing SS by N
population standard deviance (σ): the square root of the population variance
samples:
inferential stats tries to find quantities that can be generalized to the population as a whole; however, samples are generally less variable than the populations they’re trying to describe. the basic reason why is that you’re essentially more likely to have outliers in the general population than a sample regardless of how representative a sample you manage to gather. this means sample variability gives a biased estimate of population variability. that said, the bias in sample variability is consistent and predictable, so we can correct for it!
the math:
SS is identical except for minor changes in notation to reflect it being a sample’s SS instead of a population’s SS: .
(also has a computational formula, . also identical but for the lower- instead of upper-case n.
it really only changes at this step, where sample variance, denoted as s2, is defined as . then, sample standard deviation, identified by s, is just the square root of the variance, or .
sample variance (s2): the mean squared distance from the mean; obtained by dividing SS by n-1
sample standard deviation (s): equals the square root of the sample variance
by dividing by a smaller number, you have a larger result, meaning it’s a more accurate and unbiased estimator of population variance.
note: we often call the sample standard deviation the estimated population standard deviation and the sample variance the estimated population variance.
degrees of freedom (df): for a sample of n scores, df=n-1; determines the number of scores in the sample that are independent and free to vary. in other words, if you have a set of 3 scores with a mean of 5, only the first 2 can be any values—the third score can only be whatever number will allow the mean to equal 5—so it has a df of 2. note that this is the same n-1 that we use in the above equation—what we’re taking into account in the equation is the fact that we DO have that one score that can’t vary! with a population, you can find the deviation for each score by measuring its distance from the population mean, but with a sample, the value of μ is unknown, so we have to measure distances from the sample mean, which varies from sample to sample (meaning we have to compute the sample mean before we can begin conjuring deviations).
note that even taking the precaution of the df, a single calculation is never gonna be 100% accurate. instead, our most accurate estimate is the average of all the estimated population standard deviations that we calculate from all the studies that we do! if unbiased, that grand average should be equal to the population parameter.
unbiased: describes a sample statistic where the average value of the statistic is equal to the population parameter
biased: describes a sample statistic where the average value of the statistic either under- or overestimates the corresponding population parameter
note: if you calculate one (1) statistic from a sample, whether it’s the same as the population parameter or not, it’s neither biased nor unbiased—it’s simply half-baked! you have to calculate all the possible statistics and average them for it to be meaningful. only that final score can be considered biased or unbiased—the calculations in between don’t matter.
on graphs, to signal the mean and standard deviation, you draw a vertical line and label the mean, then a horizontal line/arrow extending from the mean to about halfway between it and the most extreme scores, labeled with the standard distribution (either LC sigma or s = #).
adding a constant to your values does not affect the standard distribution. multiplying your values by a constant also multiplies your standard distribution by that same constant.
adding a constant to your values adds the same constant to your mean. multiplying your values by a constant also multiplies your mean by that same constant.
in literature, SD is often used to denote a sample’s standard deviation.
note that “as a rule of thumb, roughly 70% of the scores in a distribution are located within a distance of one standard deviation from the mean, and almost all of the scores (roughly 95%) are within 2 standard deviations of the mean.” “two standard deviations” means the standard deviation times 2.
low variability = easy to find patterns in your data
error variance: in inferential statistics, the variance that exists in a set of sample data; term used to indicate that the sample variance represents unexplained or uncontrolled differences between scores. as error variance increases, difficulty finding patterns also increases.