Chapter 4: Variability

How to calculate variance

1. find the sum of the squared deviations, SS 2. Variance is the mean squared deviation (sigma)^2 = SS/N if only (df = n-1) of scores are free to vary, variance is s^2 = SS/n-1 = SS/df

The Empirical Rule

- 99.7% of data observed following a normal distribution lies within 3 standard deviations of the mean. - 68% of the data falls within one standard deviation - 95% percent within two standard deviations - 99.7% within three standard deviations from the mean.

Changing numbers effect on standard deviation

- adding constant value to every score in a distribution does not change the standard deviation - multiplying every score by a constant causes the standard deviation to be multiplied by the same constant

Properties of the standard deviation

- can be affected by extreme scores - relatively unaffected by sample size - stable under repeated sampling - takes all scores into account (do not pay attention to df, all scores are used in the SS) - directly related to the mean (subtract mean from each score) - if a constant is added to every score in a distribution, the standard deviation will NOT be changed (different from the mean which would change) - if each score is multiplied by a constant, the standard deviation will be multiplied by the same constant

Disadvantage of using range

- doesn't take all scores into account - its affected by extreme scores - fluctuates with sample size (smaller the size, larger chance of change) - can't be computed in an open-ended distribution (5, 6, 9, 14, 21 or more ] do not know the largest score) - Has no direct relationship with any other statistical measure

Disadvantages of interquartile range

- has no direct relationship with any other statistical measure - does not take all scores into account

Advantages of using interquartile range

- less affected by extreme scores - relatively unaffected by sample size - can be computed in an open-ended distribution and in ordinal data

variability

- use measures of variability to determine how spread out the scores are in a distribution.

calculating range

- use the highest real limit - lowest real limit or - highest score - lowest score +1

What does a good measure of variability do?

1. indicates how accurately the mean describes the distribution of scores. - related to the mean in some way 2. indicates how well a score (or group of scores) represents an entire distribution - how far any givin core is from the mean using standard deviation

How to calculate SS (definitional formula)

Really long time, do not use I. a. find the deviation (X - m (mew))) for each score b. square each deviation c. add the squared deviations SS = ∑(x-m)^2 II. for when the mean is not a whole number a. SS = ∑X^2 - ((∑X)^2/N)

What is the standard deviation for the following population of scores? 1, 3, 7, 4, 5

Mean = 4 2.2

What is the value of SS, sum of squared deviations, for the following population of N = 4 scores? 1, 4, 6, 1

Mean: 3 SS = ∑X^2 - ((∑X)^2/N): 54 - (12^2/4) = 18 b. 18

Interquartile range

Q3-Q1 - distance covered by the middle 50% of the distribution

How to calculate SS (computational formula)

SS = ∑X^2 - ((∑X)^2/N) (square each score and add them) - ((add each score and scare the sum)/N)

What is the variance for the following sample of n = 4 scores? Scores; 2, 5, 1, 2

SS: 9 Variance: 3

Variance calculation differences between sample and population

Sample: s^2 = SS/(n-1 <- df) Population: sigma^2 = SS/N

Variability: samples and populations

Variability changes the shape of a distribution - when the population variability is small, all of the scores are clustered close together and any individual score will necessarily provide a good representation of the entire set. - when variability is large and scores are widely spread, it is easy for one or two extreme scores to give a distorted picture of the general population

Standard deviation

a computed measure of how much scores vary around the mean score - standard distance from the mean - square root of the variance - sample standard deviation: SD = √s^2 = SS/df - population standard deviation: sigma = √(sigma)^2

What is the variance of the following set of scores? 2, 2, 2, 2, 2

a. 0

Which of the following sets of scores has the greatest variability? a. 2, 3, 7, 12 b. 13, 15, 16, 17 c. 24, 25, 26, 27 d. 42, 44, 45, 46

a. 2, 3, 7, 12

A researcher selects all the possible samples with n = 3 scores from a population and computes the sample variance, dividing by n - 1, for each sample. If the population variance is sigma^2 = 6, then what is the average value for all of the sample variances? a. 6 b. greater than 6 c. less than 6 d. impossible to determine

a. 6

Which of the following values is the most reasonable estimate of the standard deviation for the set of scores in the following distribution? a. 0 b. 1 c. 3 d. 5

b. 1

How many scores in the distribution are used to compute the range? a. 1 b. 2 c. 50% d. all

b. 2

Which of the following accurately describes the concept of standard deviation? a. the average distance between one score and another b. the average distance between a score and the mean c. the total distance from the smallest score to the largest scored. d. one half of the total distance from the smallest score to the largest score

b. the average distance between a score and the mean

Each of the following is the sum of the scores for a population of N = 4. For which population would the definitional formula be a better choice than the computational formula for calculating SS? a. ∑X = 9 b. ∑X = 12 c. ∑X = 15 d. ∑X = 19

b. ∑X = 12

What is the value of SS, the sum of the squared deviations, for the following sample? Scores: 1, 4, 0, 1

c. 9

what is the range for the following set of scores? 3, 7, 9, 10, 12

c. 9 or 10 points

What is meant by a biased statistic? a. The average value for the statistic overestimates the corresponding population parameter. b. The average value for the statistic underestimates the corresponding population parameter. c. The average value for the statistic either overestimates or underestimates the cor- responding population parameter. d. The average value for the statistic is exactly equal to the corresponding popula- tion parameter.

c. The average value for the statistic either overestimates or underestimates the cor- responding population parameter.

If the sample variance is computed by dividing by n, instead of n-1, how will the obtained values be related to the corresponding population variance? a. They will consistently underestimate the population variance. b. They will consistently overestimate the population variance. c. The average value will be exactly equal to the population variance. d. The average value will be close to, but not exactly equal to, the population variance.

c. The average value will be exactly equal to the population variance.

Which of the following is an example of an unbiased statistic. a. the sample mean b. the sample variance (dividing by n 2 1) c. both the sample mean and the sample variance (dividing by n - 1) d. neither the sample mean nor the sample variance (dividing by n - 1)

c. both the sample mean and the sample variance (dividing by n - 1)

effect of large variance

can obscure patterns in the data and create a problem for inferential statistics

Variance

commons measure of spread about the mean as center - ALSO CALLED MEAN SQUARE - the mean of the squared deviations - standard deviation squared - measures the average squared distance from the mean - sample variance: s^2 - population variance: (sigma)^2 - (deviation score (subtract mean from each score))^2

Degrees of freedom

df = n-1 - the number of scores in a sample that are free to vary - ex: if you had four scores: 2, 4, 6, 8 - M = 5 - n = 4 - ∑X = 20 The first three scores (2, 4, , 6) can all vary and become 5, 5, 5 for example - the final score must be a number that maintains the same M, n, and ∑X, so it would have to be 5 in this case

How do the mean and standard deviation work together?

knowing these two values should allow you to create a reasonably accurate image of the distribution - also allows you to describe the relative location of any individual score - mean: center of distribution - standard deviation: average distance from the mean

three measures of variability

range, variance, standard deviation

How to calculate standard deviation

square root of the variance 1. find variance 2. √SS/N

range

the distance covered by the set of scores, from smallest to largest - determined by the two extreme scores - a crude measure of variability (not great, easily affected by outliers)

Purpose of variability

to measure and describe the degree to which the scores in a distribution are spread out or clustered together - measures of qualifying variability