Chapter 5 - Notes

Range as a Measure of Variability - Range = ? Characteristics of the Range - The range describes the ______ between the _____ and ______ scores from a distribution, taking into account their respective _____. Any change in either the highest or lowest score will ______. - Because only 2 scores determine the range, the range is a _______ measure of variability. Moreover, the larger the size of the sample, the more likely the distribution could contain an ______. - The virtue of the range is that it is ______, but because of its ________, it is not a frequently reported measure of variability.

- Xhighest URL -- Xlowest LRL - overall spread... highest and lowest... real limits... affect the range. - relatively unstable... outlier. - easy to calculate... lack of stability.

For an interval or ratio scale of measurement, what measure of central tendency and measure of variability should you use for a skewed unimodal distribution?

Measure of central tendency: Mdn Measure of variability: Range

For an ordinal scale of measurement, what measure of central tendency and measure of variability should you use for a skewed unimodal distribution?

Measure of central tendency: Mdn Measure of variability: Range

For an ordinal scale of measurement, what measure of central tendency and measure of variability should you use for an approximately symmetrical unimodal distribution?

Measure of central tendency: Mdn Measure of variability: Range

- For a nominal scale of measurement, what measure of central tendency and measure of variability should you use for an approximately symmetrical unimodal distribution?

Measure of central tendency: Mode Measure of variability: None

For a nominal scale of measurement, what measure of central tendency and measure of variability should you use for a skewed unimodal distribution?

Measure of central tendency: Mode Measure of variability: None

For an interval or ratio scale of measurement, what measure of central tendency and measure of variability should you use for an approximately symmetrical unimodal distribution?

Measure of central tendency: X-bar Measure of variability: s

Estimated Population Variance - The estimated population variance, often called variance, is represented by s^2 and defined as... s^2 = ? - What are the 4 steps to calculate this? - When we try to estimate the variability of any population from a sample, the scores in the sample will usually be ______ than the scores in the _______, and we will likely _____ the ________ of the population. - It is for this reason that we don't calculate the estimated population variance by dividing the SS by ______. Though using this kind of formula will give a measure of the variance of a ______, it will provide a _______ that will always ______ the actual ______. - Though you may see a variance calculated using N rather than N-1, this variance is used as a ________ to capture ________ and not to estimate a population variance. For most purposes, we calculate the variance of a sample to provide an _______. Thus, the bias is _______ by using N-1 instead of N as the denominator in the formula for s^2.

- (The sum of (X - mu)^2) / N - 1 Or, s^2 = SS / N - 1 - Subtract the mean from each score. Square each resulting difference. Add all the squared difference values. Divide the sum by N-1. - less variable... population... underestimate the variability. - N... sample... biased estimate... underestimate... population variance. - descriptive statistic... what is going on in the sample... estimated population variance... corrected.

Measures of Variability About the Sample Mean - A more stable measure of variability would take into account _______ in the distribution. The two most commonly used measures of variability, the _____ and _______, do just that by using all of the scores to calculate the variability about the _____. - Although each of the individual deviations will provide an indication of how far an individual score is from the mean, the ________ provides no information on the variability of the distribution because it will always be _____. This problem can be avoided by ______ each deviation value before summing the deviations and obtaining the _______. - The SS is useful in obtaining a measure of how much the scores in a distribution ______, but there is a problem. In general, unless all the scores in a distribution are the _____ and thus _____ to the mean of the distribution, the value of SS will _______ as the _______ in the distribution gets larger. - In general, the more scores in a data set, the larger the sum of squares will be. We can overcome this problem by finding the _______ in a data set. This step of finding the average sum of squares will produce ________.

- all of the scores... variance and standard deviation... mean. - sum of deviations... zero... squaring... sum of squared deviations. - differ from the mean... same... equal... increase... number of scores. - average of the sum of squares. - variance.

- Knowing a measure of central tendency alone does not indicate _____ or ______ there is among the scores in the distribution. Hence, in addition to understanding measures of central tendency that provide us with information about what the typical data are, we must also understand ______ the data are. We turn to _______. - The more the scores differ from each other, the more they will also differ from the _______ and the ________ there is in the distribution. - There is ____ variability in a data set if all scores are the same. Consequently, each measure of central tendency ______ the scores in the distribution. - There is _____ variability among scores when the scores are clustered closely around the measure of central tendency. Thus, each measure of central tendency is _______ of all individuals in this group since there is _______. - There is _____ variability among the scores when the scores differ considerably from each other. Clearly, any measure of central tendency for this set of scores is _______ of the scores. - As with measures of central tendency, the _______ must be quantified.

- how much dispersion or variability... how varied... measures of variability. - measure of central tendency... more variability. - no... perfectly represent. - little... reasonably representative... little variability. - high... not as representative. - variability of scores.

The Choice of Descriptive Statistics - Researchers usually select only _____ measure of central tendency and variability to summarize their data from a study. - The choice of which descriptive statistic to use to summarize data depends principally on 3 considerations, what are they?

- one. - (1) the scale of measurement represented by the scores, (2) the shape of the frequency distribution of the scores, and (3) the intended use of the descriptive statistics for further statistical analysis.

Population Standard Deviation - If N is equal to the ______ and the ______ is known, then the population standard deviation, sigma, is given by... Sigma = ? - Mu is typically ______ and the value of sigma is usually estimated from the _____.

- population... population mean... square root of sigma^2 = the square root of (the sum of (X-mu)^2) / Npopulation - unknown... sample.

What is a Large Amount of Variability? No simple rule is used to determine whether a value for any measure of variability reflects a small or large amount of variability in a distribution of scores. Whether the variability is extensive or not is ______ to the possible range of scores that can be obtained on the variable and the mean of the scores. Journal Presentation of Measures of Variability - The _________ is the measure of variability commonly presented in journal presentations of research using _____.

- relative - estimated population standard deviation... SD.

Population Variance - The population variance is represented by _____. - If we know ____ and _____ in the population, then we can find the population variance by... Sigma^2 = ? - In actual research a population would be much _____, and it usually is _____ to measure all members of a population; thus ____ is usually not known. Consequently, sigma^2 is typically not calculated from a population, but is _____ from _______.

- sigma^2 - mu and each score... (The sum of (X - mu)^2) / Npopulation. - larger... impractical... mu... estimated from a sample.

Estimated Population Standard Deviation - Standard deviation = s = ? - How do we calculate the E.P.S.D.?

- square root of SS/N-1 - Subtract the mean from each score. Square each resulting difference. Add all the squared difference values. Divide the sum by N-1. Take the square root of the resulting quotient.

Interpreting the Variance and the Standard Deviation - In published reports of research, the _______ is presented as a descriptive statistic more often than the ______. ----- One reason for this greater use is that, because the standard deviation is the square root of the variance, it provides a measure of the variability of scores using the scores' _______. - The standard deviation provides a measure of the __________. If all the scores in a distribution are equal to the mean, then s will equal ____. - If you use ______ to predict any of the scores in the distribution, there will be zero error in your prediction. If you see a value of s equal to zero, then you know that all scores in the distribution are _______, and the mean is a ______ of all the scores. - If at least one score in the distribution is not equal to the mean, then s will take on a value ______. And the value of s will _____ as the variability of the scores ______. - A second interpretation of s involves describing scores in terms of ________ of a distribution. For many distributions, a majority of the scores are within _______ of the mean. - If the distribution is _______, then knowledge of the standard deviation provides even more information about the scores in the distribution. - We can _______ the number of the majority of responses/behaviors by finding the values that are _______ below and above the mean.

- standard deviation... variance. ----- original units. - average of how much the scores in a distribution differ from the mean... 0. - X-bar... equal to the sample mean... perfect predictor. - greater than zero... increase... increases. - how many standard deviations they are away from the mean... one standard deviation. - normally shaped. - anticipate... one standard deviation.

Further Data Analysis - The choice of measures of central tendency and variability is also determined by whether the descriptive statistics are to be used for ______. - For inferential purposes, the _____, ______, and ______ are used because they are related to the _______ of the normal distribution. - Behavioral scientists may use the ____ and _____ to summarize their data when considerations of ____ and ______ might dictate against it. - The issue of making inferences from the data sometimes overrides these other considerations. For this reason, you will find that the ____ and _____ are the most frequently used descriptive statistics in behavioral research.

- statistical inference... - mean, estimated population standard deviation, and estimated population variation... known characteristics. - sample mean and standard deviation... level of measurement and shape of the frequency distribution. - mean and the standard deviation.

Computational Formulas - These definitional formulas for the variance and standard deviation best convey conceptually ____ and _____ are, and they are easy to use with _____ data sets. When a large number of scores are involved, the definitional formulas become more cumbersome and prone to ______. Then, it is often easier to calculate s^2 or s with _____ or ______.

- what s^2 and s... small... arithmetic errors... computer-based statistical programs or hand-held calculators.