Question A

Categorise the following data types into Nominal, Ordinal, Interval or Ratio data:

SOLUTION:

1. Floor area - RATIO
2. Memory card capacity - RATIO
3. Premier league positions - ORDINAL
4. Shapes - NOMINAL
5. Latitude - INTERVAL
6. Pay grades - ORDINAL
7. Salary - RATIO
8. Playing cards - NOMINAL
9. Longitude - ORDINAL (it’s very nearly INTERVAL, however longitude loops back around)
10. Olympic medals (gold, silver, bronze) - ORDINAL
11. Vegetables - NOMINAL
12. Date - INTERVAL
13. Foot size - ORDINAL
14. Musical notes - (NOMINAL if considering their names, ORDINAL if considering their octaves too, may also be INTERVAL when considered musically. Only RATIO if considering wavelengths, but these increase exponentially.)
15. Horsepower - RATIO

Question B

1. Give an example of a data set with 6 values that has the same mean, median, and mode.

   SOLUTION: e.g. \(\{3, 3, 3, 3, 3, 3\}\). Mean is 3, median is 3, and mode is 3.

2. Give an example of a data set with 5 values that has different values of mean, median, and mode.

   SOLUTION: e.g. \(\{1, 1, 2, 3, 493\}\). Mean is 100, median 2, mode is 1.

Question C

For each of the following determine the most appropriate data visualisation and draw it:

SOLUTION:

SOLUTION:

Ages of patients in a ward: {57, 64, 69, 69, 69, 74, 75, 76, 76, 77, 78, 79, 80, 81, 81, 84, 86, 87, 87, 88, 89, 89, 92, 98}

SOLUTION:

Question D

Look at the data visualisation below:

1. Give the range, median, Q1, Q3, and IQR for the weigths of dogs, cats and rats.

   SOLUTION:

   • Dogs:
     • Range = 15
     • Q1 = 15
     • Median = 18
     • Q3 = 20
     • IQR = 5
   • Cats:
     • Range = 10
     • Q1 = 10
     • Median = 15
     • Q3 = 17
     • IQR = 7
   • Rats:
     • Range = 4
     • Q1 = 3
     • Median = 4
     • Q3 = 5
     • IQR = 2

2. Can we select a dog that is heavier than a cat?

   SOLUTION: Yes

3. Can we select a cat that is heavier than a dog?

   SOLUTION: Yes

4. Can we select a rat that is heavier than a dog?

   SOLUTION: No

5. Can we select a cat that is lighter than a rat?

   SOLUTION: No

6. A dog is selected at random, what is the probability that there exists a cat heavier than it?

   SOLUTION: 0.5. As 50% of all dogs are heavier than any can (median dog is equal to maximum cat), then there is a probability 0.5 of selecting a dog heavier than the heaviest cat, and a probability 0.5 of selecting a dog which is lighter than the heaviest cat.

7. Look at the data visualisation below:

Give a data set with 11 values that would correspond to this.

SOLUTION: For example \(\{1, 1, 3, 4, 5, 5, 5, 5, 6, 7, 7\}\), any set of values where the 1st value is 1, the 3rd is 3, the 6th is 5, the 9th is 6, and the 11th is 7.

Question E

In one day, the total amount of money spent at 2 shops by customers are shown below:

• Shop 1: {£3.00, £2.50, £7.80, £10.11, £0.99, £1.20, £3.00, £2.99, £2.40, £0.50, £5.00, £3.35, £2.40, £2.40, £5.10, £1.20, £3.00, £3.00}
• Shop 2: {£50.10, £48.30, £41.99, £50.30, £61.00, £48.20, £51.50, £39.00, £60.00, £42.12, £39.00, £55.00, £59.99, £47.80, £39.00}

1. Calculate the mean and median of each data set.

   SOLUTION:

   • Mean of Shop 1: £3.33
   • Median of Shop 1: £2.995
   • Mean of Shop 2: £48.88667
   • Median of Shop 2: £48.3

2. Calculate the variance and standard deviations of each data set.

   SOLUTION:

   • Variance of Shop 1: £5.4591
   • Standard deviation of Shop 1: £2.3365
   • Variance of Shop 2: £ 55.3188
   • Standard deviation of Shop 2: £7.4377

3. Which data set has the largest relative variance?

   SOLUTION:

   • Coefficient of variance for Shop 1: £2.3365 / £3.33 = 0.701646
   • Coefficient of variance for Shop 2: £7.4377 / £48.88667 = 0.152141
   • Therefore Shop 1 has the largest relative variance.

4. Draw box and whisker plots for both data sets on different axes and comment on their shape.

   SOLUTION:

   

   

   Shop 1 has a larger skewness and kurtosis.

Question F

1. Put the following in order of skew:

   

   SOLUTION:

   

2. Put the following in order of kurtosis:

   

   SOLUTION: