Lecture 6- Measures of Variability

Question 1

What are three measures of central tendency often used in statistics?

Answer 1

• Mean (average)
• Median (mid point)
• Mode (most common value)

Question 2

What is the aim of measuring central tendency?

Answer 2

Summarizes a data set with a single value that is representative of the data set

Question 3

In terms of measuring central tendency what occurs when the distribution is approximately normal?

Answer 3

• The mean, median and mode are approximately the same

- Therefore the mean can be used to summarize the data as no information will be ‘missed’ in doing so

Question 4

What is the purpose of measuring variability in a data set?

Answer 4

Looks at the extent to which things are not the same i.e. how scores differ from each other

Question 5

What are the two major roles of measuring variability?

Answer 5

1) Inferential statistics - the amount of variability affects the kinds of statements we can make about our population and the degree of confidence we can have in those statements. The more variability the lower our confidence should be in our trend
2) Descriptive statistics- variability is an interesting property of a data set in itself

Question 6

Why can’t we just rely on measures of central tendency to summaries data?

Answer 6

Because although two sets of data may average to have the exact same mean the distribution of the data may be completely different. For example, you could have a graph that is very narrow (all data is clustered tightly around the mean) or alternatively data could be spread out with long tails.

Question 7

What’s the difference between a low variability and high variability data set?

Answer 7

Low variability= most data clustered around mean (narrow graph)
High variability= data spread out along the scale (long tails= broad graph)

Question 8

What is the simplest way to measure variability?

Answer 8

Range.
The largest score in your data - the smallest score.
This shows that all your data points sit within this number

Question 9

Where can range fall down?

Answer 9

It ignores distribution of the data. You can have two graphs that have both the same mean and range but that are still complete different.

Question 10

What’s an alternative measure of variability to range?

Answer 10

Variance (S2)

It is calculated by taking the sum of the squared deviations from the mean and dividing it by the number of scores

Question 11

How do we get standard deviation from variance?

Answer 11

Square root it.
Variance= S^2
SD= S

Question 12

What is standard deviation?

Answer 12

• Approximately the average distance of the scores in a data set from their mean
• It is the most useful and most common measure of variability

Question 13

How does S^2 compare to the true population variance (on average)? How do we overcome this?

Answer 13

S^2 is a little bit smaller than the population variance on average. we take this into account by dividing the sum of the squared differences in our mean by n-1 not n

Question 14

What is the inflection point in a normal distribution?

Answer 14

Point where the curve starts bending outward more (start of the bend)

Question 15

Where is the inflection point in terms of the mean in a normal distribution?

Answer 15

A point of inflection will always be 1 SD from the mean

Question 16

Where is the inflection point in terms of the mean in a normal distribution?

Answer 16

A point of inflection will always be 1 SD from the mean

Question 17

What can be said about the distribution of data in a normal distribution using SDs?

Answer 17

• About 68% of all scores are within +/- 1 SD of the mean
• Nearly 96% are within +/- 2 SD of the mean
• 99.7% are within +/- 3 SD of the mean

Question 18

How do you calculate a Z score?

Answer 18

z = (data point you are interested in- mean) / standard deviation (s)

Question 19

What does a Z score tell us?

Answer 19

How far away a score is from the mean, taking into account the
variability of the scores in the data set

It allows us interpret a score in terms of its…
a) relative standing
b) variability (S)
by comparing it to a standard normal curve

Question 20

What do we mean by being able to determine a score’s relative standard?

Answer 20

Determining what proportion of scores are less than or

greater than the individual score

Question 21

What are the steps to determining an individual scores relative standard in a distribution curve?

Answer 21

1) Calculate z-score
2) Draw a picture
3) Look up value of z in z-table to find the
relative position of xi

Question 22

Given this data calculate the z score…

Assume mean family income = $37,000,
S = $5,000, value interested in is 30,000

Answer 22

z = 30,000 30,000 – 37,000/ 5,000
= – 7,000/ 5,000
z = – 1.4

Question 23

If you get a negative z score how do you look it up in a z table?

Answer 23

By using the positive entry. The bell curve is symmetrical so this doesn’t matter.

Question 24

For any normal distribution, the z-distribution obtained by converting
every data element to its z-score has what properties?

Answer 24

1) It is normal
2) mean = 0
3) S = 1

Question 25

In addition to finding the relative position of a data element in its distribution what can z scores also allow and why?

Answer 25

They can be used to compare scores from different distributions
Normally it is hard to do this because distributions vary in both their means and standard deviations, however, because z scores can transform any data into a standardized version it allows comparison