Numerical Measures

Question 1

What is the formula for the mean of a data set?

Answer 1

(Σx)/n where x represents the values of data and n is the number of values

Question 2

How is the median value found for non-grouped data?

Answer 2

(n+1)/2

Question 3

How is the median value found for grouped data?

Answer 3

n/2

(If you get an n^th value ending in .5, work out the mean between the value in front and value behind to get the value the median corresponds too)

Question 4

What is the mode?

Answer 4

The mode is the most common value in a data set.

Note for grouped data, there is a modal class. Which is defined as the class in which the modal value is contained.
Also note, not all samples have a mode.

Question 5

When working out the mean, median, mode or quartiles what information do you need?

Answer 5

You need a cumulative frequency column

Question 6

What is the range?

Answer 6

The range is the difference between the highest and lowest value in a data set

Question 7

What is the lower quartile?

Answer 7

The data at the 25th percentile of the sample.

For non-grouped data, the n^th value that represents the lower quartile is found by 0.25(n+1) where n is the cumulative frequency

For grouped data, the n^th value that represents the lower quartile is found by 0.25(n) where n is the cumulative frequency

(if you get an n^th value ending in .5 work out the mean between the value in front and the value behind to get the value the lower quartile corresponds to)

Question 8

What is the interquartile range?

Answer 8

The difference between the upper and lower quartiles

(Q₃ - Q₁)

(this is a value)

Question 9

What is the upper quartile?

Answer 9

The data value at the 75th percentile of the sample.

For non grouped data, the n^th value that represents the upper quartile is found by 0.75(n+1) where n is the cumulative frequency

For grouped data, the n^th value that represents the upper quartile is found by 0.75(n) where n is the cumulative frequency

(if you get an n^th value ending in .5 work out the mean between the value in front and the value behind to get the value the upper quartile corresponds to)

Question 10

How are variance and standard deviation related?

Answer 10

Variance = Standard Deviation ²

Question 11

What actually is variance?

Answer 11

A measure of how far each data point squared is from the mean, and therefore represents the spread of the data

Question 12

How is variance found?

Answer 12

• Find the mean of the data points
• Calculate the difference between each data point and the mean value (write this as a new list of values)
• Square the difference between each data point and the mean
• Find the sum of your new list of values
• Write the final answer as the relevant unit squared

Question 13

How do you find the variance, SD, median and quartiles with your calculator?

Answer 13

• MENU (6)
• 1-Variable (1)
• Enter data and frequency
• AC
• OPTN
• 1-Variable Calc (2)

For grouped data, find the midpoint of the class and put it into the calculator

Note do not use cumulative frequency in the calculator

Question 14

How do you deal with grouped data when inputting into the calculator to find numerical measures?

Answer 14

Use the midpoint of the data as the value to input

Question 15

What is grouped and non-grouped data?

Answer 15

Grouped data refers to data given in class intervals (e.g 10-20)

Non-grouped data refers to individual pieces of data (e.g 6,24,69,420)

Question 16

How can you convert grouped data into non-grouped data?

Answer 16

Write out the heading of the group as many times as the frequency states

(e.g a group of 3 people with 4 cats each,
becomes, 4,4,4)

Note this also works in reverse

Question 17

When there are gaps in a continuous grouped data set (lengths 0-9, 10-19, 20-29), what do you always do first?

Answer 17

Adjust class widths to the value for which they would no longer round to the original values

(0-9, 10-19) becomes (0-9.5, 9.5-10.5)

Then find the midpoint column

Question 18

When there are gaps in discrete grouped data sets (ages 0-5, 6-10, 11-15), what do you always do first?

Answer 18

Adjust class widths so that the final value of the width is the first value of the next width

(0-5, 6-10 11-15 … ) becomes (0-6, 6-11, 11-16 etc)

Then find the midpoint column

Question 19

What is continuous data?

Answer 19

Data which can take up any value (e.g girth, length and height)

Question 20

What is discrete data?

Answer 20

Data which can be counted and has finite values (e.g sausages, boys and pens)

Question 21

What is the ‘formula’ for linear interpolation?

Answer 21

(UB-LB)/(UF-LF) = (Q-LB)/(N-LF)

This basically states the proportion of the boundaries range to frequency range is the same as the proportion of the median - lowest boundary value to the median - lowest frequency

Question 22

State an assumption of linear interpolation

Answer 22

Data is evenly spread within the boundaries

Question 23

How do you find ‘N’ in linear interpolation?

Answer 23

For median n is the (cumulative frequency / 2)
For LQ n is the (cumulative frequency / 4)
For UQ n is 3 x (cumulative frequency / 4)

Question 24

What are the steps of linear interpolation?

Answer 24

– Adjust class widths of grouped data for any gaps
– Add a cumulative frequency column
– Input the cumulative frequency as n and sub into relevant equation (median / quartile)
– Find the class in which this value for n falls
– Draw interpolation diagram
– Find UB and LB by reading class width
– Find UF and LF by finding cumulative frequency on either side of the class
– Sub these values including N into the equation and solve for Q

Question 25

What is the equation for standard deviation as coded data?

Answer 25

S_y = S_x / b

Where y is coded data and x is the original data where b is a constant

Question 26

What are the advantages and disadvantages of using the median as a measure of location?

Answer 26

Advantages:
- Useful for non-numerical data
- Always an observed data value
Disadvantages:
- Affected by an outlier
- Does not use all data

Question 27

How do you draw a linear interpolation diagram?

Answer 27

• Draw a horizontal straight line

-Draw 3 vertical lines at the top, bottom and middle of your line

• Write upper and lower boundaries on the top as well as Q
• Write upper and lower frequencies on the bottom as well as the value of n (calculated by frequency equation initially)
• Solve for Q using the equation

Question 28

True or False data given in linear interpolation questions the data given is always grouped

Answer 28

True you will never be given non grouped data

Question 29

What is the equation for variance?

Answer 29

S_xx/n

or

((Σx²)/n) - x̄²)

Question 30

What is the equation for standard deviation?

Answer 30

(S_xx/n)^1/2

(((Σx²)/n) - x̄²))^1/2

Question 31

What is the general equation for coded data?

Answer 31

y = (x-a) / b

where y is the coded data value, x is the original data and a and b are constants

Question 32

What is the equation to find the coded mean?

Answer 32

ȳ = (x̄ - a)/b

Where y is the coded mean and x is the original mean and a and b are constants

Question 33

What are the advantages and disadvantages of using the mode as a measure of location?

Answer 33

Advantages:
- Not affected by an outlier
- Useful for non-numerical data

Disadvantages:
- Does not use all data
- May be multiple modes

Question 34

What are advantages and disadvantages of using the mean as a measure of location?

Answer 34

Advantages:
- Large data set makes outliers negligible
- Uses all data values

Disadvantages:
- Affected by outliers in small data sets

Question 35

When you have discrete data with gaps do you amend the gaps or not?

Answer 35

You do not amend the gaps. You only amend gaps in continuous grouped data

Question 36

What are the advantages and disadvantages of using the range as a measure of spread?

Answer 36

Advantages:
- Reflects the full data set

Disadvantages:
- Affected by outliers

Question 37

What are the advantages and disadvantages of using the interquartile range as a measure of spread?

Answer 37

Advantages:
- Not affected by outliers

Disadvantages:
- Does not reflect the full data set

Question 38

What are the advantages and disadvantages of using the standard deviation as a measure of spread?

Answer 38

Advantages:
- Outliers are negligible in large data sets

Disadvantages:
- Outliers have a big impact on small data sets

Question 39

What does sigma (Σ) notation express?

Answer 39

Sigma (Σ) refers to the ‘sum of’
For example, sigma x (Σx) means the sum of all the values of x

Question 40

What does standard deviation actually mean?

Answer 40

A measure of how far each data point is from the mean, and therefore represents the spread of the data

Question 41

Does addition/subtraction (when coding data) affect the mean and standard deviation? (skip this card)

Answer 41

Adding or subtracting will affect the mean of the data but not the standard deviation. This is because all data points have increased/decreased by the same value and so the distance from the mean is no different.

The mean will change by the same value as the addition or subtraction

Question 42

Does multiplication/division (when coding data) affect the mean and standard deviation?

Answer 42

Multiplying or dividing affects both the mean and standard deviation
The mean will change by the same factor as the division or multiplication

Question 43

What is the mean?

Answer 43

The mean is the sum of all data divided by the number of pieces of data

It is calculated in the same way for grouped and non-grouped data.

Question 44

What is the equation for standard deviation?

Answer 44

(S_xx/n)^1/2