2- Visualisation and Presentation of Data

Question 1

Discrete data

Answer 1

When the data values are quantitative and the numbers are finite or countable

Question 2

Continuous data

Answer 2

Result from infinitely many possible quantitative values, where the collection of values is not countable.

Question 3

Levels of measurement

Answer 3

Nominal
Ordinal
Interval
Ratio

Question 4

Nominal level of measurement

Answer 4

• The nominal level of measurement is characterised by data that consist of names, labels, or categories only.
• The data cannot be arranged in an ordering scheme.
• Cannot be used for calculations.
• Numbers are sometimes assigned to the different categories.

Question 5

Example of nominal level of measurement

Answer 5

Social security numbers: Substitutes for names, they do not count or measure anything
Yes/No/Undecided :survey responses

Question 6

Ordinal level of measurement

Answer 6

• Data are the ordinal level of measurement if they can be arranged in some order
• The differences (obtained by subtraction) between data values either cannot be determined or are meaningless.

Question 7

Examples of ordinal level of measurement

Answer 7

• Course grades: A college professor assigns grades of A, B, C, D or E. These grades can be arranged in order, but we cannot determine the differences.
• Ranks

Question 8

Interval level of measurement

Answer 8

• Data are at the interval level of measurement if they can be arranged in order, and differences between data values can be found and are meaningful.
• Data at this level do not have a natural zero starting point at which none of the quantity is present

Question 9

Example of Interval level of measurement

Answer 9

• Data are at the interval level of measurement if they can be arranged in order, and differences between data values can be found and are meaningful.
• Data at this level do not have a natural zero starting point at which none of the quantity is present

Question 10

Examples of Interval level of measurement

Answer 10

Outdoor temperatures:18 ℃ and 34 ℃ are examples of data at this interval level of measurement. We can determine their difference of 16 ℃, but there is no natural starting point. Though 0 ℃ seems like a starting point, it is arbitrary and does not represent the total absence of heat.
Years: The years 1492 and 1776 can be arranged in an order, and we can determine the difference, and is meaningful but, time did not begin in the year zero, so zero is arbitrary instead of being a natural zero starting point representing “no time”.

Question 11

Ration Level of measurement

Answer 11

• Data are the ratio level of measurement if they can be arranged in order, differences can be found and are meaningful, and there is a natural zero starting point.
• Here, zero indicates that none of the quantity is present
  For data at this level, differences and ratios are both meaningful.

Question 12

Examples of Ratio Level of measurement

Answer 12

• Car lengths: car lengths of 106 inch for a Smart car, 212 inch for a Mercury Grand Marquis, ( 0 inch represents no length, 212 inch is twice as long as 106 inch)
• Class times: the times of 50 mins and 100 mins for a statistics class (0 min represents no class time, and 100 mins is twice as long as 50 mins)

Question 13

Extra of ratio measurement

Answer 13

• Ratio level of measurement is called ratio because the zero starting point makes ratios meaningful.
• Consider two quantities when one number is twice the other and ask whether “twice” can be used to correctly describe the data.

Example:
- We can say a person with a hight of 6ft is twice as tall as a person with hight 3ft – the heights are the ratio level of measurements
However, we cannot say 50 ℃ is twice as hot as 25 ℃, temperatures are not at the ratio level

Question 14

Tables which can represent qualitative data

Answer 14

Frequency Table
Relative frequency table
Percentage frequency table
Cumulative frequency table

Question 15

Tables which can represent quantitative data

Answer 15

Frequency, Relative frequency, percentage frequency, cumulative frequency tables – instead of using category names we use discrete values here

Question 16

Charts which can represent quantitative data

Answer 16

Histogram
Bar chart
Line graph
Scatter graph
Box plot

Question 17

Charts which can represent qualitative data

Answer 17

Bar graph
Pie graph

Question 18

Measures of location

Answer 18

The mean,
The median,
The mode,
Skewness and kurtosis

Question 19

Measures of dispersion (variability)

Answer 19

The range and Percentile
Quartiles and interquartile range
The mean deviation
The variance
The standard deviation.

Question 20

The mean

Answer 20

• Perhaps the most important measure of location is the mean, or average value, for a variable.
• If sample data, the mean is denoted by 𝑥̅
• If a population, the Greek letter 𝜇 is used to denote the mean.

Question 21

Mode- when there is no mode?

Answer 21

Unimodal

Question 22

Mode- when there is 2 modes?

Answer 22

Bimodal

Question 23

Skewness definition

Answer 23

A measure of the degree of asymmetry of a distribution.

Question 24

Kurtosis definition

Answer 24

A measure of whether the data are peaked or flat relative to a normal distribution.

Question 25

Description of negative skew

Answer 25

Has a high frequency or relatively high values and a low frequency of relatively low values, so the mean is dragged toward the left (the low values) of the distribution.

Question 26

Mean, median and mode of negative skew

Answer 26

mode > median > mean

Question 27

Description of normal skew

Answer 27

Said to be symmetrical ; the mean, median and mode have the same value and thus coincide at the same point of the distribution.

Question 28

Mean, median and mode of normal skew

Answer 28

Mean=mode=median

Question 29

Description of positive skew

Answer 29

Has a high frequency of relatively low values and low frequency of relatively high values, so the mean is dragged toward the right (the high values) of the distribution

Question 30

Mean, mode and median of positive skew

Answer 30

Mean > median > mode

Question 31

Is the mean a good measure of central tendancy with skewed data?

Answer 31

No- because it is sensitive to extreme values.

Question 32

Kurtosis definition

Answer 32

A measure of whether the data are peaked or flat relative to a normal distribution.

Question 33

Positive kurtosis

Answer 33

More peaked amongst the three distribution

Question 34

Positive kurtosis

Answer 34

More peaked amongst the three distributions- Leptokurtic

Question 35

Normal kurtosis

Answer 35

Mesokurtic

Question 36

Negative kurtosis

Answer 36

Flattest distribution- no peak/ obvious curve- Platykurtic

Question 37

Mean value, standard deviation and kurtosis of all kurtosis’

Answer 37

The mean value (therefore the standard deviation and the variance for all three distributions are the same).

Question 38

Percentile

Answer 38

A percentile provides information about how the data are spread over the interval from the smallest value to the largest value.
The 𝑝^𝑡ℎ percentile is a value such that at least 𝑝 per cent of the observations are less than or equal to this value and at least (100- 𝑝) per cent of the observations are greater than or equal to this value.

Question 39

How to calculate the pth percentile

Answer 39

1- Arrange the data in ascending order (smallest value to largest value)
2- Compute an index 𝑖, 𝑖=𝑝/100 𝑛
where 𝑝 is the percentile of interest and 𝑛 is the number of observations.
3 a) If 𝑖 is not an integer, round up. The next integer greater than i denotes the position of
the pth percentile.
b) If 𝑖 is an integer, the 𝑝^𝑡ℎ percentile is the average of the values in positions 𝑖 and 𝑖+1

Question 40

Variance definition

Answer 40

The variance is a measure of variability that uses all the data. The variance is based on the difference between the value of each data and the mean. The difference is called a deviation about the mean.

Question 41

How is the deviation about the mean expressed for a sample?

Answer 41

(𝑥_𝑖−𝑥̅)

Question 42

How is the deviation about the mean expressed for the population?

Answer 42

(𝑥_𝑖−𝜇)

Question 43

Standard deviation definition

Answer 43

The standard deviation is defined to be the positive square root of the variance.

Question 44

What does a low standard deviation indicate?

Answer 44

Low standard deviation indicates that the values tend to be close to the mean of the data set.

Question 45

What does a high standard deviation indicate?

Answer 45

High standard deviation indicates that the values are spread out over a wider range

Question 46

Histogram frequency equation

Answer 46

Frequency= class interval x frequency density