Chapter 9 - Collecting, Representing and Interpreting Data

Question 1

What are the 6 Sampling Techniques?

Answer 1

Sampling Techniques:

• Simple Random Sampling
• Systematic Sampling
• Stratified Sampling
• Opportunity Sampling
• Quota Sampling
• Cluster Sampling

Question 2

Define Population and Sample.

Answer 2

A population is the set of things that data is going to be collected about. A sample is a subset of that population.

Question 3

Define a Parameter in Statistics, and therefore, define a statistic.

Answer 3

A parameter is a number that describes the entire population. A statistic is a number taken from a single sample to estimate the parameter.

Question 4

Define a sample bias.

Answer 4

A sampling method is biased if it creates a sample that does not represent the population.

Question 5

How do you find the mode of a set?

Answer 5

The mode of a set of data is the value or category that occurs most often or has the largest frequency. For grouped data, the modal interval or modal group is normally given.

Question 6

How do you find the mean of a set?

Answer 6

To work out the mean x̄ of a set of n observations, calculate their sum (of the x values) and divide the result by n.

x̄ = ∑x / n

x̄ = ∑fx/∑f, where f is frequency.

Question 7

Define Systematic Sampling.

Answer 7

• Systematic Sampling - Find a sample of size n from a population size N by taking one member from the first k members of the population at random, and then selecting every kth member after that, where k = N/n.

Question 8

Define Stratified Sampling.

Answer 8

• Stratified Sampling - When you know you want distinct groups to be represented in your sample, split the population into these distinct groups and then sample within each group in proportion to its size.

Question 9

Define Opportunity Sampling.

Answer 9

• Opportunity Sampling -Take samples from members of the population you have access to until have a sample of the desired size.

Question 10

Define Quota Sampling.

Answer 10

• Quota Sampling - When you know want distinct groups to be represented in your sample, decide how many members of each group you wish to sample in advance and use opportunity sampling until you have a large enough sample for each group.

Question 11

Define Cluster Sampling.

Answer 11

• Cluster Sampling - Split the population into clusters that you expect to be similar to each other, then, take a sample from these clusters.

Question 12

Define Simple Random Sampling.

Answer 12

• Simple Random Sampling - Every Member of a population is equally likely to be chosen. I.E, assign each member of the population with a number. Then, choose a random number.

Question 13

What are the four measures of spread?

Answer 13

The four measures of spread are:

• Range
• Interquartile Range
• Variance
• Standard Deviation

Question 14

Define Range.

Answer 14

The range of a set of data is the largest value minus the smallest value.

Question 15

Define the Medium, and how to calculate it.

Answer 15

The median of a set of data is the middle value of data listed in order of size.

To calculate the position of the medium of a set of n observations, work out the value of (n+1)/2. If this value is a whole number, the medium is in that position. If it is decimal, the median is the mean of the two positions on either side.

Question 16

How do you evaluate the lower quartile and the upper quartile?

Answer 16

• To evaluate the lower quartile for a set of n ordered values, work out the value of (n+1)/4.
• To evaluate the upper quartile for a set of n ordered values, work out the value of 3(n+1)/4.
• To find the position, use the above formulas in the same manner as finding the medium.

Question 17

How do you find the interquartile range?

Answer 17

To find the interquartile range, subtract the value of the upper quartile from the value of the lower quartile.

Question 18

Define population variance, and how to calculate it.

Answer 18

The variance of a set of data measures how spread out the values are from the mean.

𝞂² = ∑(x-µ)² / n

Question 19

Define population standard deviation.

Answer 19

The population standard deviation is the square root of the population variance.

𝞂 = √𝞂² = √(∑(x-µ)² / n)

Question 20

Define sample variance, and how to calculate it.

Answer 20

An unbiased estimate of population variance using a sample of n observations with sample mean x̄ is given by the sample variance, s².

s² = ∑(x-x̄)² / n-1
or
s² = (∑x² / n-1) - (∑x)²/n(n-1)

Question 21

Define sample standard deviation.

Answer 21

The sample standard deviation is the square root of the sample variance.

s = √s²

Question 22

What are the pros and cons of using the mode as a measure of central tendency?

Answer 22

Pros of mode:
- Useful for non-numerical data.
- Not usually affected by outliers.
- Not usually affected by errors or omissions.
- Is always an observed data point.
Cons of mode:
- Doesn’t use all of the data.
- It may not be representative if it has a low frequency.
- There may be other values with similar frequency.

Question 23

What are the pros and cons of using the median as a measure of central tendency?

Answer 23

Pros of median:
- Not affected by outliers.
- Not significantly affected by errors.
Cons of median:
- Doesn't make use of all the data.

Question 24

What are the pros and cons of using the mean as a measure of central tendency?

Answer 24

Pros of mean:
- When the data set is very large a few extreme values have negligible impact.
Cons of mean:
- When the data set is small a few extreme values or errors have a big impact.

Question 25

What are the pros and cons of using the range as a measure of spread?

Answer 25

Pros of range:
- Reflects the full data set.
Cons of range:
- Distorted by outliers.

Question 26

What are the pros and cons of using the IQR as a measure of spread?

Answer 26

Pros of IQR:
- Not distorted by outliers.
Cons of IQR:
- Does not reflect the full data set.

Question 27

What are the pros and cons of using the standard deviation as a measure of spread?

Answer 27

Pros of standard deviation:
- When the data set is very large a few outliers have negligible impact.
Cons of standard deviation:
- When the data set is small a few outliers have a big impact.

Question 28

What is the five number summary? And how can it be plotted?

Answer 28

Data is often summarised by five main statistics.

The five number summary gives the minimum value, lower quartile, median, upper quartile and maximum value.

These can be plotted using a box-and-whisker (or box) plot.

Question 29

What are the methods for plotting continuous single variable data?

Answer 29

The methods for plotting continuous single variable 
data:
- Box Plot
- Cumulative Frequency Graph
- Histogram

Question 30

How do you plot a cumulative frequency graph?

Answer 30

For a cumulative frequency graph, the x-coordinates are the upper boundary of each interval, and the y coordinates are the sum of the frequencies up to those points.

Question 31

How do you plot a histogram?

Answer 31

For a histogram, the width of the rectangles are the size of the intervals, and the height is the frequency density, where:
Frequency Density = Frequency ÷ Class Width

Question 32

What are the three types of correlation?

Answer 32

The three types of correlation:

• Perfect Positive Correlation, r = +1
• Perfect Negative Correlation, r = -1
• No Correlation, r = 0.

Question 33

What are the rules of correlation? (for whether variables affect one another)

Answer 33

When a change in one variable does affect the other, they have a causal connection.

Correlation without a causal connection is known as spurious correlation.

Question 34

What are the advantages and disadvantages of: Box Plots

Answer 34

Question 35

What are the advantages and disadvantages of: Histograms

Answer 35

Question 36

What are the advantages and disadvantages of: Cumulative Frequency curves?

Answer 36