Central Tendency, Variability, & Z-scores

Question 1

What data classes are best for mean?

Answer 1

interval, ratio

Question 2

What data classes are best for mode?

Answer 2

nominal, ordinal, interval, ratio

Question 3

What data classes are best for median?

Answer 3

ordinal, interval, ratio

Question 4

Under what conditions might a median be a better measure of central tendency than the mean?

Answer 4

• when the data is ordinal (mean does not apply)
• interval/ratio data if there are extreme values

Question 5

It should seem clear how the mean and the median are measures of the central tendency of the data since the mean is is a familiar average and the median is the middle. However, explain why mode is also considered a measure of central tendency?

Answer 5

most data sets peak in the middle (bell shape). The mode is the highest frequency so it’s usually in the middle somewhere.

Question 6

For any data set, what is Σ(X − X ̅)?

Answer 6

Σ(X − X) may be written as:
ΣX − ΣX ̅ = ΣX − nX ̅ = ΣX − n(ΣX)/n = ΣX − ΣX = 0.

Question 7

The following data represent a sample of the time to complete a certain task in minutes and seconds (mm:ss).

6:30, 11:15, 6:22, 11:32, 8:12, 5:02, 9:17, 6:51, 8:44, 7:45, 9:37, 7:28, 4:29, 7:42

compute the mean:
compute the std. dev.:

Answer 7

Since the values are given in minutes and seconds they first need to be converted to either minutes and decimal parts (eg. 6:30 = 6 + 30/60 = 6 + 0.5000 = 6.5000min) or to seconds (eg. 6:30 = 6*60 + 30 = 360 + 30 = 390s) so that they can be easily added.

mean: 7:55
std. dev.: 2:04

Question 8

For a certain set of data, the mean and standard deviation are computed.

How does X ̅ (data treated as sample) compare to μ (data treated as a population)?

How does s (data treated as sample) compare to σ (data treated as a population)?

Answer 8

X ̅ is the sample mean, μ is the population mean; they are calculated the same way.

standard deviation of sample (N-1) vs. population (N)

Question 9

Given the following sample data set:

        6, 12, 9, 7, 8, 4, 3, 12, 15 

Compute the mean.
What is the median?
What is the mode?
Compute the variance.
Compute the standard deviation.

Answer 9

mean: 8.44
median: 8
mode:12
variance: 15.77
standard deviation: 3.97

Question 10

For the following sample data set:
X frequency
52 5
54 8
57 2

Compute the mean.
Compute the variance.
Compute the standard deviation.

Answer 10

mean: 53.73
variance: 2.635
standard deviation: 1.62

Question 11

The following sample data of the number of communications are taken from logs of communication with Distance Education students:

5, 9, 5, 23, 27, 55, 34, 7, 30, 15, 22, 60, 14, 52, 297, 8, 51, 15, 51, 35, 15, 39, 137, 43, 38, 14, 93, 7 

Compute the mean.
Compute the standard deviation.
Draw a boxplot with the minimum, Q1, Q2, Q3, and maximum.
Which is a better representation of the central tendency: mean or median? Explain.

Answer 11

mean: 42.89
std. dev.: 57.28
Minimum: 5
Q1: 14
Q2: 28.5
Q3: 51
Maximum: 297

The mean is; this is due to extreme values.

Question 12

If the two largest values in the sample data set of the previous problem were omitted,

Compute the mean.
Compute the standard deviation.
Draw a boxplot with the minimum, Q1, Q2, Q3, and maximum.
Which is a better representation of the central tendency: mean or median? Explain.

Answer 12

mean: 29.50
std. dev.: 21.68
minimum: 5
Q1: 14
Q2: 25
Q3: 43
Maximum: 93

Mean may now be a better measure because extreme outliers have been removed.

Question 13

Consider the following data set:
21, 34, 18, 26, 30, 35, 24, 29, 25

If this is a population, compute the mean.
If this is a sample, compute the mean.
If this is a population, compute the standard deviation.
If this a sample, compute the standard deviation.

Answer 13

μ=26.9
X ̅= 26.9
σ= 5.34
s= 5.67

Question 14

If we had a set of ordinal values (not interval/ratio), could you create a boxplot?

Answer 14

Technically yes, because quartiles depend only on the position in the ordered data set. Thus, one could determine the positions in the ordered set for Q1, Q2 (median), and Q3 and the first and last position for the minimum and maximum. However, without interval/ratio data, visualizing this with a boxplot would not make sense.

For example, imagine you ask 9 people what size drink they ordered, small, medium, or large. The ordered data might be: small, small, small, medium, medium, large, large, large, large. Q1 is position 2.5 (small), Q2 is position 5 (medium), and Q3 is position 7.5 (large), minimum is position 1 (small) and maximum is position 9 (large).

Question 15

Typically we consider quantitative data that is symmetric about the mean. If we have a data set that has a few extreme high values, then

a. How is it skewed?
b. Would you use a mean or median? Why?

Answer 15

It is positively skewed (right-skewed)

You would use median since it is less sensitive to extreme values.

Question 16

14. For the MCAT, µ = 500 and σ = 10. What is the probability of an individual getting a score greater than 502.5?

Answer 16

z=0.25
p=0.413

Question 17

15. For the MCAT, µ = 500 and σ = 10.

What is the minimum score would you have to obtain to be in the top 5%?

What is the minimum score you would have to obtain to be in the top 2.5%?

Answer 17

95%
500 + (1.64 x 10) = 516.4

97.5%
500 + (1.96 x 10)= 519.6

Question 18

Correlational method

Answer 18

looking for relationships between variables (correlation or regression)

Question 19

Experimental method

Answer 19

manipulating one variable to determine if this causes changes in another variable

Question 20

independent variable

Answer 20

what we control/manipulate

Question 21

dependent variable

Answer 21

what we measure (is influenced)

Question 22

confounding/extraneous variables

Answer 22

other things impacting (things impacting dependent that aren’t independent)

Question 23

random assignment

Answer 23

equal chance to end up in group (bigger the better)

helps decrease extraneous variables

Question 24

experimental vs control groups

Answer 24

experimental: at least 2 different groups
control: group with no treatment (placebo)

Question 25

Placebo

Answer 25

any treatment that has no active properties

Question 26

hypothetical constructs

Answer 26

an explanatory variable which is not directly observable

we must find ways to operationalize these

Question 27

operational definition

Answer 27

how do we assign a number?

Question 28

population

Answer 28

all the people we want to apply results to (we control/decide)

Question 29

sample

Answer 29

we find a subset of the pop. that is representative of the whole pop. (random)

Question 30

random sample

Answer 30

random group from population

Question 31

descriptive statistics

Answer 31

summarizes data

Question 32

inferential statistics

Answer 32

trying to infer back to population (generalize)

Question 33

parameter vs. statistics

Answer 33

statistics for sample; parameter for population

Question 34

sampling error

Answer 34

the difference between stat. from sample and it’s parameter

Question 35

discrete vs. continuous variable

Answer 35

discrete: categories w/ nothing in between

continuous: infinite values between any two categories

Question 36

quantitative vs. categorical data

Answer 36

quantitative: directly measuring something (continuous data)

categorical data: counts of things (discrete variables)

Question 37

scales of measurements

Answer 37

nominal: no inherent order of different categories (weakest)

ordinal: one group is above other, not evenly spaced (can use median)

interval: there’s equal interval, but no true zero

ratio: there’s equal interval, but is true zero

Question 38

frequency distributions

Answer 38

• real lower limit
• real upper limit
• midpoint

Question 39

visualizing data

Answer 39

• histogram: frequency distribution turned into a graph. We can see shape o destitution and the spread of the data.
• line graph: good for looking at change/time
• scatterplot: tells us about relationships between variables. shows pos. & neg. relationships. strength of relationship based on how linear.
• boxplot: box represents 50% of data.

Question 40

shapes of distributions

Answer 40

symmetrical
- unimodal: bell-shaped (normal dist.)
- bimodal: clear 2 peaks (one can be higher)
- rectangular: data of equal freq. for all values

asymmetrical
- pos. skewed: skewed to right (not norm. but unimodal)
- neg. skewed: skewed to left (not norm. nut unimodal)

Question 41

central tendency

Answer 41

mean: avg. of all numbers
median: middle number in list
mode: most freq. number

Question 42

variability
-range
-interquartile range
- variance
- standard dev.

Answer 42

range: x(max)- x(min)
interquartile range:
Q1=.25 x (# in data data set)
Q2=.50 x (# in data data set)
Q3=.75 x (# in data data set)
IQR= Q3-Q1
variance: avg. squared dev. of each number from mean
std. dev.: sqrt var. (takes away squared unit)

Question 43

z-scores

Answer 43

raw score to z-score: x=u+2o
z-score to raw score: z=(x-u)/o

Question 44

standardize a dist.

Answer 44

shape of standard distribution: the shape of the distribution of z-scores will be the same as the shape of the original dist. raw scores
mean: z-score dist. always have mean of zero so and above=+ and any below=-
standard deviation: the z-score dist. will always have a standard dist. of 1. The numerical val. of z-score is exactly the same number of standard deviation from the mean

Question 45

normal distribution

Answer 45

empirical rule: the following apprrox. holds
- 68% of obs. fall between u-o & u+o
95% of obs. fall between u-2o & u+2o
99.7% of obs. fall between u-3o & u+3o

Question 46

un-biased stat

Answer 46

a statistic whose long range average is equal to the parameter it estimates