Chapter 4: Variability

How many scores are free to vary in a sample?

All scores except one are free to vary

What are the characteristics of standard deviation?

Always positive, affected by the value of every score in a distribution, almost always reported with the mean

Biased estimator

Any sample statistic obtained from a randomly selected sample that does not equal the value of its respective population parameter, its value will be less than the population variance

Unbiased estimator

Any sample statistic obtained from a randomly selected sample that equals that value of its respective population parameter, its value will equal the population variance

Sample standard deviation

Calculated by taking square root of sample variance

Population standard deviation

Calculated by taking square root of the population variance

Empirical rule

Data normally distributed, 99.7% of data lie within 3 SD, 95% of data lie within 2 SD, and 68% of data lie within 1 SD of the mean

Range

Difference between the largest value and smallest value in a data set

Deviation

Difference of each score from its mean

A researcher reports that participants in her study had a mean age of 21.3 years, with a range equal to 3, based on this info, is it possible that one of the participants was 25 years old in her study?

No, the range is too small

The empirical rule is stated for data with what type of distribution?

Normal distribution

Degrees of freedom (for sample variance)

Number of scores in a sample that are free to vary, all scores except one are free to vary in a sample (n-1)

A researcher selects a population of eight scores where SS = 72. What is the population variance in this example?

Population variance = 9

When data is divided into 4 equal parts, what is the data split into?

Quartiles

A researcher collects the following scores: 1, 2, 3, 4, 5, 6, 7, 8. What is the range of these scores

Range = 8 - 1 = 7

Interquartile range (IQR)

Range of values between upper (Q3) and lower (Q1) quartiles

The population variance is 121, what is the standard deviation?

SD = 11

The sample variance is 121, what is the standard deviation?

SD = 11

A researcher measure the following scores: 12, 14, 16, 18, 20. What is the SS for the data?

SS = 40

Describe the same of squares (SS) in words

SS is the sum of the squared deviations of scores from their mean

A researcher measure the following data: 3, 3, 3, 4, 4, 4. What is the sample variance?

Sample variance = 0.3

Sum of squares (SS)

Some of squared deviations of scores from their mean, SS is the numerator in the variance formula

Medain

Splits data in half

Deciles

Splits data into 10 parts

Percentiles

Splits data into 100 parts

Quartiles

Splits data into 4 parts

Standard deviation

Square root of variance, measure of variability for average distance that scores deviate from their mean

How do you compute standard deviation?

Take the square root of the variance

How is the computational formula different from the definitional formula for variance?

The computational formula does not require that you compute the mean to compute SS in the numerator

How does calculating the sample variance differ from calculating the population variance?

The denominator for sample variance is (n-1), not N

What is the formula for computing the range?

The largest value minus the smallest value in a data set

The standard deviation is a measure used to determine the average distance that each score deviates from what?

The mean

Median quartile

The median value of a data set at the 50th percentile

Lower quartile

The median value of the lower half of a data set at the 25th percentile

Upper quartile

The median value of the upper half of data at the 75th percentile

An instructor measures the following quiz scores: 6, 8, 7, 9 (SD = 1.29). If the instructor subtracts two points from each quiz score, how will the value for the standard deviation change?

The standard deviation will not change (SD = 1.29)

True or False: the sample variance is unbiased when dividing SS by (n-1)

True

True or False: when all scores in a population are the same, the variance will always be equal to 0

True

True or False: with identical data sets, the definitional and computational formula for sample variance will always produce the same solution, give or take rounding error.

True

Why do we square each deviation in the numerator of variance?

We want to compute how far scares from their mean without ending up with a solution equal to zero every time

A researcher measures the following sample of scores (n = 3): 1, 4, 7. (a) use the definitional formula to calculate variance, (b) use the computational formula to calculate variance, (c) are the answers the same?

(a) definitional variance = 9 (b) computational variance = 9 (c) yes

How many standard deviations from the mean will contain at least 99% of data for any type of distribution?

+/- 10 SD

A researcher records the number of times that 10 students ugh during a final exam, he records the following data: 0, 0, 0, 3, 3, 5, 5, 7, 8, 11. True or False: in this example, the range will be smaller than the interquartile range?

False, the range will be larger than the IQR

Measures used to divide data into two or more parts

Fractiles

Why is the variance a preferred measure of variability?

It includes all scores in its computation

Semi-interquartile range (SIQR)

Measure of half the distance between the upper quartile (Q3) and lower quartile (Q1), computed by dividing IQR in half

Variability

Measure of the dispersion or spread of scores in a distribution (around the mean) ranging from 0 to +∞

Population variance

Measure of variability computed only when all scores in a given population are recorded (SS/N)

Variance

Measure of variability for average squared distance that scores deviate from the mean

Sample variance

Measure of variability, computed when only a portion or sample of data is measure in a population (SS/n-1)

Fractiles

Measures that divide data into two or more equal parts

True or False: a scientist measures the following data: 23, 23, 23, 23, 23, 23. The value for the sample variance and population variance will be the same.

True, because the variance is 0

Measure of the dispersion or spread of scores in a distribution and ranges from 0 to +∞

Variability

Computational formula for variance

Way to calculate population and sample variance without needing to sum squared differences of scores from their mean to compute SS in the numerator

Definitional formula for variance

Way to calculate population variance and sample variance that requires summing the squared differences of scores from their mean to compute the SS in the numerator

A researcher records five scores: 3, 4, 5, 6, and x. If the mean in this distribution is 5, then what is the value for x?

x = 7