# STATS Chap 4

descriptive versions

When describing how far the scores are spread out from x̅, we use the sample variance (S²x) and the sample standard deviation (Sx). When describing how far the scores are spread out from μ, we use the population variance (σ²x) and the population standard deviation (σx).

normal distribution

Approximately 34% of the scores are between the mean and the score that is 1 standard deviation from the mean. -characteristic bell shape of any of these always places 68% of the distribution between the scores that are +1Sx and -1Sx from the mean.

squared sum of X

Calculated by adding all scores and then squaring their sum -The symbol (∑X)² indicates to find the squared sum of X. To do so, work inside the parentheses first, so find the sum of the X scores. Then square that sum.

sum of squared Xs

Calculated by squaring each score in a sample and adding the squared scores -The symbol ∑X² indicates to find the sum of the squared Xs. To do so, first square each X(each raw score) and then sum—add up—the squared Xs.

sample variance

The average of the squared deviations of scores around the sample mean symbol S²x Always include the squared sign (²). The capital S indicates that we are describing a sample, and the subscript X indicates it is a sample of X scores. Formula S²x=∑(X-x̅)²/N is important because it shows you the basis for the variance -always an unrealistically large number because we square each deviation

population variance

The average squared deviation of scores around the population mean -symbol σ²x=∑(X-μ)²/N

sample standard deviation

The square root of the sample variance; interpreted as somewhat like the "average" deviation -is as close as we come to the "average of the deviations," -we can precisely describe where most of the scores in a distribution are located. symbol Sx (the square root of the symbol for the sample variance.) to compute we first compute everything inside the square root sign to get the variance formula Sx=√[∑(X-x̅)²/N -allows us to envision how spread out the distribution is and, correspondingly, how accurately the mean summarizes the scores. -If Sx is relatively large, then a large proportion of scores are relatively far from the mean, which is why they produce a large "average" deviation. Therefore, we envision a relatively wider distribution. -If Sx is smaller, then more often scores are close to the mean and produce a smaller "average." Then we envision a narrower distribution.

Variance

only very roughly analogous to the "average" deviation -used extensively in statistics -does communicate the relative variability of scores -a number that generally communicates how variable the scores are -The larger, the more the scores are spread out

range

The distance between the highest and lowest scores in a set of data - Highest score- lowest score -measure of variability only with nominal or ordinal data. -With nominal data, we compute by counting the number of categories we're examining -With ordinal data, it is the distance between the lowest and highest rank

biased estimators

The formula for the variance or standard deviation involving a final division by N, used to describe a sample, but that tends to underestimate the population variability -the under- and overestimates do not cancel out. Instead, although the sample variance and sample standard deviation accurately describe a sample, they are too often too small to use as estimates of the population. -out of the N deviations in any sample, only N- 1 of them (the N of the sample minus 1) actually reflect the variability in the population.

unbiased estimators

The formula for the variance or standard deviation involving a final division by N- 1; calculated using sample data to estimate the population variability

population standard deviation

The square root of the population variance, or the square root of the average squared deviation of scores around the population mean -symbol σx(The is the lowercase Greek letter s, called sigma) -formula σx=√[∑(X-μ)²/N] -we can interpret the population standard deviation as the "average" deviation of the scores around μ, with 68% of the scores in the population falling between the scores that are at +1σx and -1σx from μ.

unbiased estimated population variance

The symbol is the lowercase s²x divide by the smaller quantity N- 1

estimated population standard deviation

The unbiased estimate of the population standard deviation calculated from sample data using N- 1 -simply added the square root symbol to the formula for the variance: is the square root of the estimated variance. formula sx=√[∑(X-μ)²/N-1]

estimated population variance

The unbiased estimate of the population variance calculated from sample data using - 1 formula s²x=∑(X-μ)²/N-1 finding the amount each score deviates from the mean, which will then form our estimate of how much the scores in the population deviate from μ. The only novelty is that in computing the "average" of the squared deviations, we divide by N- 1 instead of by N.

The Computing Formula for the Sample Variance Is

This says to first find the sum of X (∑X) , square that sum, and divide the squared sum by N. Then subtract that result from the sum of the squared Xs (∑X²) . Finally, divide that quantity by N.

infer the variability

When the complete population of scores is unavailable, we _____________ of the population based on a sample by computing the unbiased estimators (s²x or sx). These inferential formulas require a final division by N- 1 instead of by N.

standard deviation

allows us to quantify "around" the mean -first find the raw score that is located at "plus 1 standard deviation from the mean" or, in symbols, at +1Sx We also find the score located at "minus 1 standard deviation from the mean," or at -1Sx

variability

refers to the differences between scores, which we describe by computing the variance and standard deviation. In each, we are finding the difference between each score and the mean and then calculating something, more or less, like the average deviation. -Any standard deviation is merely the square root of the corresponding variance -compute the descriptive versions when the scores are available

measures of variability

the statistics for describing the differences among scores -Statistics that summarize the extent to which scores in a distribution differ from one another -the range, the variance, and the standard deviation.

unbiased estimated population standard deviation

the symbol is the lowercase sx divide by the smaller quantity N- 1