Deck 4: Summarizing Data: Variability

Question 1

A researcher records the following data: 4,4,4,4,and 3.How would you describe the variability of these data?

A)It is equal to zero because scores are approximately the same.
B)It is negative because 3 is less than the other scores in the distribution.
C)It is very small (close to 0)because scores are approximately the same.
D)It is very large (much greater than 0)because 3 is an outlier in the data.

Answer 1

It is very small (close to 0)because scores are approximately the same.

Question 2

A measure of the average squared distance of scores from the mean is called the

A)range
B)IQR
C)variance
D)sum of squares

Answer 2

variance

Question 3

All other things being equal,how will the value of the sample variance be different from the population variance?

A)The sample variance will always be a smaller value than the population variance.
B)The sample variance will always be a larger value than the population variance.
C)The sample variance,but not the population variance,can be zero.
D)The sample variance,but not the population variance,can be negative.

Answer 3

The sample variance will always be a larger value than the population variance.

Question 4

Regardless of the number of scores in a distribution,the range only includes ___ score(s)in its calculation.

A)one
B)two
C)at most two
D)the average

Question 5

A researcher decides to split scores on an exam into quartiles.She determines that a score of 64 is at the 25th percentile,a score of 74 is at the 50th percentile,and a score of 80 is at the 75th percentile.What is the interquartile range (IQR)for these data?

A)16
B)10
C)6
D)There is not enough information to answer this question.

Question 6

The variance and standard deviation can never be

A)zero
B)negative
C)smaller than the mean
D)larger than the mean

Question 7

A psychologist treats 16 patients and records the number of sessions required to complete a behavioral therapy treatment for each patient.She computes SS = 800.Assuming the 16 patients constitute all patients under her care (so the population of her patients),what is the standard deviation for these data?

A)53.3 sessions
B)50 sessions
C)7.1 sessions
D)7.3 sessions

Question 8

A researcher measures the time (in seconds)it takes a sample of 26 participants to respond to a stimulus presented on a computer screen.The standard deviation for response times is 6.In this example,what is the value for SS?

A)36
B)150
C)900
D)There is not enough information to answer this question.

Question 9

A researcher records the following scores for an Olympic gymnast following her routine: 9.9,9.8,9.6,9.5,9.7,9.1,8.9,and 9.8.What is the range for the scores?

A)1.0 (9.9 to 8.9)
B)0.3 (9.8 to 9.5)
C)0.5 (9.6 to 9.1)
D)It is not possible to compute a range with an even number of scores.

Question 10

A researcher records the time in seconds it takes a sample of participants to walk alone through a dark portion of campus.The researcher computes SS = 1,200.Assuming that a sample of 25 participants was observed in this study,what is the standard deviation for these data?

A)48 seconds
B)50 seconds
C)6.9 seconds
D)7.1 seconds

Question 11

Fractiles are ways that data can be divided into two or more equal parts.An example of a fractile includes

A)the median
B)quartiles
C)percentiles
D)all of the above

Question 12

A researcher records the number of individual instruments identified by students listening to a piece of classical music.He splits the data into quartiles and reports that students correctly identified 3 instruments at the 25th percentile,5 instruments at the 50th percentile,and 8 instruments at the 75th percentile.What are the IQR and SIQR for these data?

A)the IQR is 3;the SIQR is 1.5
B)the IQR is 2;the SIQR is 1.0
C)the IQR is 5;the SIQR is 2.5
D)There is not enough information to answer this question.

Question 13

Which of the following is an example of a distribution with NO variability?

A)scores: 3,5,4,4,5,and 3
B)scores: 3,3,3,3,33,and 3
C)scores: 1,2,3,4,5,and 6
D)scores: 5,5,5,5,5,and 5

Question 14

An interquartile range removes

A)the top and bottom 25% of scores in a distribution before calculating range
B)scores above only the 75th percentile before calculating range
C)scores below only the 25th percentile before calculating range
D)all of the above

Question 15

Measures of variability can range in value from

A)- $\infty$ to + $\infty$
B)0 to +
C)It depends on whether the variability is positive or negative.
D)It depends on the value of the mean for a given distribution.

Question 16

The sum of the squared deviations of scores from their mean

A)is computed the same for samples and populations
B)is computed by squaring each deviation to avoid a zero solution in the numerator
C)is the numerator for the sample variance and population variance
D)all of the above

Question 17

Which of the following describes the definitional formula for variance?

A)It is stated in terms of how variance is defined.
B)It is computed using SS in the numerator.
C)It is a measure of the average squared distance that scores deviate from their mean.
D)all of the above

Question 18

The range,a measure of variability,

A)is the difference between the largest (L)and smallest (S)value in a list of scores
B)is the most informative when used to describe data sets without outliers
C)includes only two values in its computation,regardless of the number of scores in a distribution
D)all of the above

Question 19

The SIQR can be defined as

A)the range of a distribution divided in half
B)the IQR of a distribution plus the range
C)a measure of half the distance between and

D)all of the above

Question 20

Which of the following distributions has the largest variability?

A)scores: 1,3,6,9,and 12
B)scores: 22,24,26,28,and 30
C)scores: 2,3,4,5,and 6
D)scores: 1,2,3,4,and 5

Question 21

Why is it important to divide by df to compute sample variance?

A)doing so makes the sample variance an unbiased estimator of the population variance
B)doing so ensures that the value for sample variance varies between 0 and 1
C)doing so minimizes the likelihood that the sample variance will be a negative value
D)it is not important;dividing by df is optional

Question 22

A researcher selects a sample of 24 participants and has them complete a survey on dating preferences.In this example,what are the degrees of freedom for sample variance and what does this figure represent?

A)df = 24;it represents the number of scores that are free to vary in a sample.
B)df = 23;it represents the number of scores that are free to vary in a sample.
C)df = 23;dividing SS by df makes the sample variance a biased estimator of the population variance.
D)df = 24;dividing SS by df makes the sample variance an unbiased estimator of the population variance.

Question 23

A researcher computes the definitional formula for SS,as finds that ∑(x-M)= 112.If this is a sample of 20 scores,then what would the value of population variance be using the computational formula?

A)5.6
B)5.9
C)112
D)not possible to know because the scores are not given

Question 24

The sample variance is computed by dividing SS by ____;whereas the population variance is computed by dividing SS by ____.

A)N;df
B)df;n - 1
C)df;N
D)n - 1;df

Question 25

A researcher computes the definitional formula for SS,as finds that ∑(x-M)= 44.If this is a sample of 12 scores,then what would the value of sample variance be using the computational formula?

A)3.7
B)4.0
C)44
D)not possible to know because the scores are not given

Question 26

A researcher measures the number of trials it takes two samples of participants to master a new task.In both samples,SS = 240.Sample A consisted of 12 participants and Sample B consisted of 18 participants.Which sample is associated with the largest variance?

A)Sample A
B)Sample B
C)Both samples have the same variance.
D)There is not enough information to answer this question.

Question 27

Which of the following values is NOT needed to compute sample variance using the computational formula,but is needed to compute the definitional formula?

A)sample mean
B)scores (x)
C)sample size
D)degrees of freedom

Question 28

A researcher computes the computational formula for SS,as finds that ∑x = 22 and ∑x² = 126.If this is a sample of 4 scores,then what would SS equal using the definitional formula?

A)4
B)5
C)104
D)not possible to know because the sample mean is not given

Question 29

Which of the following is true about the computational formula for variance?

A)The computational formula will always produce the same solution as the definitional formula (give or take rounding errors).
B)It is a short-cut method for calculating variance when the population or sample size is large.
C)It is derived mathematically from the definitional formula.
D)all of the above

Question 30

A researcher computes the computational formula for SS,as finds that ∑x = 39 and ∑x² = 271.If this is a sample of 6 scores,then what would SS equal using the definitional formula?

A)17.5
B)3.5
C)232
D)not possible to know because the sample mean is not given

Question 31

The advantage of squaring the deviation of each score from the mean and then summing is that

A)it makes the sample variance an unbiased estimator of the population variance
B)it makes the degrees of freedom for sample variance equal to n - 1
C)it produces a minimal positive solution that is not zero,so long as all scores are not exactly the same value
D)both A and C

Question 32

The sample variance is:

A)an unbiased estimator of the population variance
B)associated with n - 1 degrees of freedom
C)computed by dividing SS by df
D)all of the above

Question 33

The definitional formula ______ the computational formula for SS.

A)estimates
B)explains
C)diminishes
D)equals

Question 34

How is the sample variance computed differently from the population variance?

A)only one formula includes a computation for SS
B)the calculation in the numerator is different
C)the calculation in the denominator is different
D)both B and C

Question 35

All other things being equal (so assuming that the value of SS never changes),as sample size increases,

A)the degrees of freedom for sample variance decrease
B)the value of sample variance decreases
C)the value in the numerator for sample variance increases
D)the value in the denominator for sample variance decreases

Question 36

A reason for squaring deviations to compute SS in the numerator includes which of the following?

A)The sum of the differences of scores from their mean is zero.
B)The sum of the squared differences of scores from their mean is minimal.
C)Squaring scores can be corrected easily by square rooting.
D)all of the above

Question 37

A(n)________ is a sample statistic that equals a population parameter on average.

A)biased estimator
B)degrees of freedom
C)unbiased estimator
D)sum of squares

Question 38

The degrees of freedom for the sample variance

A)are equal to the sample size
B)are equal to the sample size minus one
C)can vary between - $\infty$ and +
D)both B and C

Question 39

Each deviation in the numerator for variance is squared because

A)without squaring each deviation,the solution for SS would be zero
B)this inflates the value for variance,making it more accurate
C)without squaring each deviation,the solution could be negative
D)both A and C

Question 40

A researcher records a sample of 30 exam scores and finds that the population mean for these data is larger than the sample variance.Is this possible?

A)yes,the population variance is usually larger than the sample variance
B)no,the population variance will only be larger when at least 50 scores are recorded
C)no,the sample variance will be larger because SS is divided by degrees of freedom in the denominator of the variance formula
D)not possible to know because the scores are not given

Question 41

A researcher records the following scores for attention during a video game task for two samples.Which sample has the largest standard deviation? Sample A: 10,12,14,16,and 18
Sample B: 20,24,28,32,and 36

A)Sample A
B)Sample B
C)Both samples have the same standard deviation.

Question 42

A researcher reports that the time (in minutes)it takes children who are "picky eaters" to finish their vegetables is negatively skewed,with children finishing their vegetables in 4.2±1.0 (M±SD)minutes.Based on Chebyshev's theorem,we can conclude that

A)68% of children finished their vegetables in 3.2 to 5.2 minutes.
B)50% of children finished their vegetables in at least 3.2 minutes.
C)99.7% of children finished their vegetables in 2.2 to 6.2 minutes.
D)At least 89% of children finished their vegetables in 1.2 to 7.2 minutes.

Question 43

All other things being equal,the value in the numerator is computed the same for sample variance and population variance.

Question 44

A main disadvantage of computing a range is that it takes into consideration only two values regardless of the number of scores in a distribution.

Question 45

Which of the following is the best explanation for why the standard deviation is almost always reported with the mean?

A)The standard deviation measures the spread of scores from the mean,so it is important to know both the mean and the standard deviation.
B)The standard deviation is also a measure of central tendency,so it is important to report this value with the mean.
C)The mean and standard deviation estimate basically the same thing,so these values are typically reported together.
D)This is not true;the standard deviation is rarely reported with the mean.

Question 46

Suppose a researcher selects a sample of 16 participants and computes SS = 60.In this example,the standard deviation is 4.

Question 47

Suppose a researcher selects a sample of 19 participants and computes SS = 162.In this example,the standard deviation is 3.

Question 48

Which of the following is a property of the standard deviation?

A)The standard deviation varies between - $\infty$ and + .
B)The standard deviation is used to describe qualitative variables.
C)Multiplying each score times the same constant will change the standard deviation by that constant.
D)Adding the same constant to each score will change the standard deviation by that constant.

Question 49

The 50th percentile is also called the median quartile.

Question 50

As a general rule,an interquartile range is a better estimate of variability than the IQR.

Question 51

A researcher computes the following scores: 12,8,14,12,and 10.If these scores were substituted into each formula,the sample variance would be smaller than the population variance.

Question 52

A researcher records the following motor assessment scores for two samples of athletes.Which sample has the largest standard deviation? Sample A: 8,10,12,15,and 18
Sample B: 16,18,20,23,and 26

A)Sample A
B)Sample B
C)Both samples have the same standard deviation.

Question 53

The definitional formula for variance is stated in terms of how variance is defined.

Question 54

A researcher records the following scores on a working memory quiz for two samples.Which sample has the largest standard deviation? Sample A: 2,3,4,5,6,7,and 8
Sample B: 4,5,6,7,8,9,and 10

A)Sample A
B)Sample B
C)Both samples have the same standard deviation.

Question 55

The interquartile range and semi-interquartile range are the same measure.

Question 56

A researcher records the sound (in decibels)during a series of lessons taught by a substitute teacher at a local elementary school.In his study,he found that the sound was 80±6 (M±SD)decibels.Assuming the data are normally distributed,which of the following is an appropriate conclusion?

A)68% of classes were between 68 and 80 decibels.
B)5% of classes were louder than 68 decibels.
C)95% of classes were between 68 and 92 decibels.
D)all of the above

Question 57

A main disadvantage of computing a range is that it is a simple and quick measure of variability.

Question 58

A researcher records the number of classroom interruptions during each of two class sessions.Which session has the largest standard deviation? Session A: 12,15,18,24,and 30
Session B: 8,10,12,16,and 20

A)Sample A
B)Sample B
C)Both samples have the same standard deviation.

Question 59

Which of the following is consistent with the empirical rule?

A)68% of all scores lie within one standard deviation of the mean.
B)95% of all scores lie within two standard deviations of the mean.
C)99.7% of all scores lie within three standard deviations of the mean.
D)all of the above

Question 60

A researcher measures the amount of coffee consumed by college students while studying during the final exam week.In her study,she found that students drink 2.3±0.8 (M±SD)cups of coffee per study session.Assuming the data are normally distributed,which of the following is the most appropriate conclusion?

A)The average student drinks less than 2.3 cups of coffee per study session.
B)Most students drink between 1.5 and 3.1 cups of coffee per study session.
C)Most students drink between 2.3 and 3.9 cups of coffee per study session.
D)Most students drink between 0.7 and 2.3 cups of coffee per study session.

Question 61

Adding or subtracting the same constant to each score will not change the standard deviation.

Question 62

The computational formula for the variance requires that you know the value of the mean.

Question 63

For a set of data that is normally distributed with 32±4 (M±SD),we know that 68% of the data will fall between scores of 28 and 36.

Question 64

The empirical rule states that all scores will fall within 3 standard deviations of the mean.

Question 65

The sample standard deviation is equal to 6 when SS = 360 and n = 11.

Question 66

For nonnormal distributions,99.7% of data will fall within three standard deviations of the mean.

Question 67

When the sample variance is computed as SS divided by n,it will underestimate the population variance on average.

Question 68

Suppose a researcher observes a population of 25 people and computes SS = 360.In this example,the variance is 15.

Question 69

The sample variance is an unbiased estimator of the population variance when we divide SS by df.

Question 70

If SS = 50 for a data set of 10 scores in a population,then the standard deviation is equal to 5.0.

Question 71

The population standard deviation is equal to 10 when SS = 100 and n = 10.

Question 72

The degrees of freedom for sample variance is equal to N.

Question 73

One reason the denominator for sample variance is (n - 1)is because this makes it an unbiased estimator of the population variance.

Question 74

Suppose a researcher observes a population of 12 people and computes SS = 192.In this example,the standard deviation is 4.

Question 75

The standard deviation is typically reported in research reports with the mean for a data set.

Question 76

The standard deviation is always positive or equal to 0.

Question 77

If all scores in a distribution are negative values,then the standard deviation will also be a negative value.

Question 78

Multiplying or dividing each score using the same constant will not change the standard deviation.

Question 79

The sample variance will be larger than the population variance for the following data: 3,5,4,7,3,5,4,8,and 9.

Question 80

On average,the sample variance equals the population variance only when dividing SS by df.