Exploring Data - Topic 3: Numerical Summaries

Question 1

What are the major features that can be summed numerically?

Answer 1

Minimum
Maximum
Centre (mean, median)
Spread (standard deviation, range, IQR)

Question 2

What are numerical summaries? Advantages and disadvantages?

Answer 2

Numerical summaries are a collection of measures that try to describe as much as possible about the data set in as few numbers as possible.

An advantage is that it allows for easy communication and comparisons

A disadvantage is that it loses a lot of info when doing so

Question 3

What is the mean?

Answer 3

It is the average of ALL the data. It is the point where the data is balanced

I.e. the higher readings and lower readings all cancel each other out

Question 4

How do you calculate mean?

Answer 4

Mean = Sum of the data / size of the data

Question 5

What is the median?

Answer 5

It is the middle datapoint when the data is ordered from smallest to largest

Question 6

How do you calculate median?

Answer 6

For odd sized data: It is the middle number (i.e. (n+1)/2)

For even sized data: It is the average of the 2 middle points

Question 7

What situations would a median be preferred over a mean?

Answer 7

Within datasets with high variation and outliers. This is because the mean will be easily influenced by these other variables, and cause a very different mean.

Thus, medians are used when there are lots of outliers as it provides a better measure of the centre than the mean (as that one is influenced by outliers a lot)

Question 8

What does robustness/robust mean? Give an example

Answer 8

It is when something is a good summary for skewed data as it isn’t affected by outliers

I.e. the median is said to be robust and is a good summary for skewed data as it isn’t affected by outliers

Question 9

What is a left skewed dataset?

Answer 9

Also known as a negatively skewed dataset, the tail is headed towards the negative side.

Here, we expect the mean to be smaller than the median

Question 10

What is a right skewed dataset?

Answer 10

Also known as a positively skewed dataset, the tail is headed towards the positive side

Here, we expect the median to be smaller than the mean

Question 11

Which meaasure of central tendenc is the best for describing a centre which is skewed?

Answer 11

Median

Mean for a symmetrical distribution

Question 12

Can there be cases when neither mean or median are helpful in describing centre?

Answer 12

Yes. This typically occurs with bimodal data

Question 13

What are limitations of the mean?

Answer 13

Doesn’t account for outliers which could influence the mean

Potential for misinterpretation

Market could have changed

Question 14

What is a limitation of the median?

Answer 14

Not accounting for all data points

Question 15

What is the wrong way to measure spread?

Answer 15

We could calculate the gap between each data value and the mean, and then average these gaps. However, the average will always be zero because from the definition, the mean gap must be zero, as the mean is the balancing point of the gaps

Question 16

Thus, what is the new, better way to measure spread?

Answer 16

A better alternative to calculating the mean of the data minus mean is through standard deviation. This involves the RMS (Root, mean, square)

The RMS measures the avg set of numbers, regardless of sign

Question 17

What are the steps to RMS?

Answer 17

Square the numbers (Each data point - mean) , mean the result, then root it

Question 18

What does standard deviation population measure and what is the equation to measure it?

Answer 18

Measures the s.d. of the population

Root ((Sum of (x - average) ^2) / N )

Question 19

What does standard deviation sample measure, and what is the equation to measure it?

Answer 19

Measures the s.d. of the sample

Root ((Sum of (x - average)^2 / (N-1))

Question 20

How do you know the differnece of when to use sample and population s.d.?

Answer 20

Population SD is when you have every set of data available, sample SD is when you don’t have every set of data available (with reference to the research question)

It is very much dependent on what the research is trying to achieve

For example;
assume ‘data’ =Newtown property prices during April - June 2017

if just looking at ‘Newtown Property Prices during April-June 2017’, then the ‘data’ is the whole population

If we want to look at all property prices/ prices in general, then the ‘data’ is considered a sample

Question 21

What % of data lies within what s.d. from the mean

Answer 21

68% of data –> 1 s.d. from the mean

95% of data –> 2 s.d. from the mean

99.7% of data –> 3 s.d. from the mean

Question 22

What are standard units?

Answer 22

They are also known as z scores, and determines how many standard deviations a score is above or below the mean

To compare2 data points, we can compare with reference to standard units

Question 23

How is a standard unit (z score calculated)

Answer 23

(Data point - mean) / SD

Question 24

What is the Interquartile range (IQR)?

Answer 24

It is the range of the middle 50% of the data. More formally, IQR = Q3 - Q1, where Q1 is the 25th percentile and Q3 is the 75th percentile

Question 25

What are quartiles?

Answer 25

Quartiles divide the data into quarters

Question 26

What is the mathematical reasoning for what an outlier is?

Answer 26

The lower and upper thresholds is both a distance of 1.5IQR from the Q3 or Q1

Lower threshold = Q1 - 1.5IQR

Upper threshold = Q3 + 1.5IQR

Question 27

What are the pairs of measures of central tendency and variability that we have to do?

Answer 27

Mean and standard deviation

Median and IQR

Question 28

What is coefficient of variation? (CV)

Answer 28

It is used to compare data dispersion between distinct series of data

Question 29

WHat is the formula for coefficient of variation?

Answer 29

SD / Mean

Question 30

What are examples of what CV is used for?

Answer 30

Analytical chemistry

Engineering + physics

Economics for determining volatility of securities