Statistics ch 3: Vive la Différence
Variability
Reflects how scores differ from each other. Can also be called spread or dispersion. A measure of how different scores are from one another. More accurate: how different scores are from a particular score. Instead of comparing each score to every other score in the distribution, the one score that could be used as a comparison is the MEAN. A measure of how much each score in a group differs from the mean.
Three measures of variability
1. Range 2. Standard deviation 3. Variance
As the size of the sample gets larger/increases (and moves closer to the population in size), the n-1 adjustment has far less impact on the difference between the biased and unbiased estimates of the standard deviation.
All other things being equal, then, the larger the size of the sample, the less difference there is between the biased and unbiased estimates of the standard deviation.
Standard deviation
As a measure of variability: it tells us how much each score in a set of scores, on the average, varies from the mean. Practical applications: can be used to help us compare scores from different distributions, even when the means and standard deviations are different.
4. Why does the standard deviation get smaller as the individuals in a group score more similarly in a test? And why would you expect the amount of variability in a measure to be relatively less with a larger number of observations than with a smaller one?
As individuals score similarly, they are closer to the mean, and the deviation about the mean is smaller. Hence, the standard deviation is smaller as well. There's strength in numbers, and the larger the data set, the more inclusive it is. That is, it includes more, rather than fewer, values that are similar to one another- hence, less variability.
Three measures of variability (Range, Standard deviation and Variance)
Commonly used to reflect: the degree of variability, spread or dispersion in a group of scores.
Average and variability
Can be used to describe the characteristics of a distribution and show how distributions differ from one another. Each is an important descriptive statistic.
Important points about the Standard deviation
Computed as the average distance from the mean. So you will need to first compute the mean as a measure of central tendency, do not fool around with the median or mode when computing standard deviation. The larger the standard deviation, the more spread out the values are, and the more different they are from one another. Just like the mean, the standard deviation is sensitive to extreme scores. When computing the standard deviation of the sample and you have extreme scores, make note of it somewhere in your report and in your data interpretation. If s=0, there is absolutely no variability in the set of scores and the scores are essentially identical in value. This will rarely happen.
5. Compute the range, the unbiased and biased standard deviations, and the variance for the following set of scores. 94, 86, 72, 69, 93, 79, 55, 88, 70, 93
Exclusive range: h-l 94-55= 39 Get mean: 79.9 Unbiased SD: √Σ(X-X¯)^2/n-1 √Σ(X-X¯)^2/n-1 (94-79.9)^2+(86-...)^2+(72-...)^2+.../10-1 1,544.9/9 then square root everything SD=13.10 Variance= 13.10^2 = 171.66 Biased SD: √Σ(X-X¯)^2/n 1,544.9/10 same, except I don't subtract by 1 SD= 12.42 Variance=154.49 The difference is due to dividing by a sample size of 9 (for the unbiased estimate) as compared by a sample size of 10 (for the biased estimate). You also practiced the unbiased and biased estimate of the variance.
2. How to get exclusive and inclusive ranges, when given high and low scores
Exclusive: h-l Inclusive: h-l+1
3. Why would you expect more variability on a measure of personality in college freshmen than you would on a measure of height?
For the most part, first-year students have stopped growing by that time, and the enormous variability that one sees in early childhood and adolescence has evened out. On a personality measure, however, those individual differences seem to be constant and are expressed similarly at any age.
3. Variance
It is the standard deviation but squared. Formula: s^2= Σ(X-X¯)^2/n-1 Same formula but without the square root bracket over the whole thing. If you do the standard deviation and never complete the last step (taking the square root), you have the variance. The variance equals to the standard deviation times itself (or squared) s^2= s * s
Variability
It reflects how different scores are from one another.
Summary
Measures of variability help us to even more fully understand what a distribution of data points looks like. Along with a measure of central tendency, we can use these values to -distinguish distributions from one another -effectively describe what a collection of test scores, heights or measures of personality looks like and what those individual scores represent. Now that we can think and talk about distributions (thanks to measures of central tendency and variability), let's explore ways we can look at them...
2. Standard deviation
Most frequently used measure of variability. It is a deviation from something that is standard. Can be abbreviated as s or SD Represents the average amount of variability in a set of scores. In practical terms: the average distance from the mean. The larger the standard deviation is, the larger the average distance each data point is from the mean of the distribution, and the more variable the set of scores is. Formula: s= √Σ(X-X¯)^2/n-1 Σ is sigma, which tells you to find the sum of what follows X is the individual score X¯ is the mean of all the scores n is the sample size This formula finds the difference between each individual score and the mean (X-X¯), squares each difference, and then sums them all together. Then it divides the sum by the size of the sample (minus 1) and takes the square root of the result. As you can see, the standard deviation is an average deviation of the mean.
1. Range
Most general measure of variability Gives you an idea of how apart scores are from one another. Computed by subtracting the lowest score in a distribution from the highest score in the distribution: highest minus lowest score Formula: r= h - l h is the highest score in the data set l is the lowest score in the data set Since it tells you how different the highest and lowest values in a data set are from one another, then the range shows how much spread there is from the lowest to the highest point in a distribution. Range is fine as a general indicator of variability, but should not be used to reach any conclusions about how individual scores differ from one another.
Standard deviation and Variance
Similarities: Both are measures of variability, dispersion and spread. The formulas used to compute them are very similar. You see them reported all over the place in the "results" sections of journals (mostly standard deviation) Differences: The standard deviation is stated in the original units from which it was derived (because we take the square root of the average summed squared deviation) The variation is stated in units that are squared (because the square root of the final value is never taken)
1. Why is the range the most convenient measure of dispersion, yet the most imprecise measure of variability? When would you use the range?
The range is the most convenient measure of dispersion because it requires only that you subtract one number (the lowest value) from another number (the highest value). It's imprecise because it does not take into account the values that fall between the highest and lowest values in a distribution. Use the range when you want a very gross (not very precise) estimate of the variability in a distribution.
Mean deviation/ Mean absolute deviation
The sum of the absolute values of the deviations from the mean (which you get by finding the difference between every actual score and the mean(X-X¯)) divided by the number of scores.
6. In #5 just above, why is the unbiased estimate greater than the biased estimate?
The unbiased estimate is always larger than the biased estimate because the unbiased estimate actually overestimates the value of the statistic intentionally to be more conservative. And the numerator (n-1) for the unbiased statistic is always less than for the biased estimate (which is n), resulting in a larger value.
Absolute value
The value regarding the sign.
4,4,4,4,4
This set of scores has no variability at all (the scores do not differ from each other). Same mean.
3,4,4,5,4
This set of scores has the same mean (4) and has less variability than the previous set.
7,6,3,3,1
This set of scores shows some variability.
When it comes to descriptive statistics and describing the characteristics of a distribution...
Two parts of the story: -Averages -Measures of variability
7. Use SPSS to compute all the descriptive statistics for the following set of three test scores over the course of a semester. Which test had the highest average score? Which test had the smallest amount of variability? they give you scores for test 1, test 2, test 3
You place all the scores from scratch, then Analyze-> descriptive statistics-> frequencies then statistics and you tick all measures of central tendency and variability Answers: Test 2 has the highest average score, and Test 1 has the smallest amount of variability. Also, there are multiple modes, and SPSS computes and reports the smallest value.
