Chapter 3: Summarising Data

Question 1

What is a measure of central tendency?

Answer 1

Represents the ‘centre’ of a set of data, including mode, median, and mean.

Question 2

Define mode in data.

Answer 2

The one that appears the most; the most common value.

Question 3

What is a modal class?

Answer 3

The class with the highest frequency.

Question 4

What is the median?

Answer 4

The middle value of a dataset.

Question 5

How do you find the median of discrete data?

Answer 5

1. Put the numbers in order from smallest to largest. 2. Find the (n+1)th value, which indicates the median position.

Question 6

What is the formula to find the median position?

Answer 6

(n + 1) / 2.

Question 7

What should you do if the median position is a decimal?

Answer 7

Find the two surrounding values and average them.

Question 8

How do you find the median in grouped data?

Answer 8

Identify the median class which contains the median position.

Question 9

What is the estimated median using linear interpolation?

Answer 9

Use ½ n to find the median position and calculate within the median class.

Question 10

What is the mean (arithmetic mean)?

Answer 10

The sum of all values divided by the number of values.

Question 11

Provide the formula for mean.

Answer 11

𝑥̅ = ∑𝑥 / n.

Question 12

How do you calculate the mean from a frequency table?

Answer 12

Add an extra column for f × x, sum it, and divide by total frequency.

Question 13

What is the formula for weighted mean?

Answer 13

Weighted Mean = ∑(weight × value) / ∑weights.

Question 14

What is the geometric mean?

Answer 14

The nth root of the product of all values.

Question 15

Why is transforming data useful?

Answer 15

To simplify calculations with large numbers.

Question 16

What happens to the mode when new values are added?

Answer 16

It could change if the new value affects which value appears most.

Question 17

How does adding a value greater than the median affect the median?

Answer 17

The median might increase.

Question 18

What is the range in statistics?

Answer 18

The difference between the largest and smallest values.

Question 19

What is the formula for range?

Answer 19

Range = Largest Value - Smallest Value.

Question 20

Define interquartile range (IQR).

Answer 20

The middle 50% of the data when in order.

Question 21

What is the formula for the interquartile range?

Answer 21

IQR = Upper Quartile - Lower Quartile.

Question 22

What is the lower quartile (LQ)?

Answer 22

The value at 25% of the way through the data.

Question 23

True or False: The mean is always affected by extreme values.

Answer 23

True.

Question 24

List the advantages of using mode.

Answer 24

• Easy to use
• Always a value in the data
• Unaffected by extreme values
• Can be used with quantitative and qualitative data

Question 25

List the disadvantages of using mode.

Answer 25

* May not exist or may have multiple modes * Cannot be used to calculate measures of spread * Not always representative of the data.

Question 26

What are the advantages of using median?

Answer 26

* Easy to find when data is in order * Unaffected by outliers * Best with skewed data * Can calculate quartiles, IQR, and skew.

Question 27

What are the disadvantages of using median?

Answer 27

* May not be a data value * Not always representative of the data.

Question 28

What are the advantages of using mean?

Answer 28

* Uses all the data * Can calculate standard deviation and skew.

Question 29

What are the disadvantages of using mean?

Answer 29

* May not be a data value * Affected by extreme values or outliers * Can be distorted by open-ended classes.

Question 30

What is the formula for Interquartile Range (IQR)?

Answer 30

IQR = UQ - LQ

Question 31

Define Lower Quartile (LQ).

Answer 31

The value ¼ of the way through the data; 25% of the data is less than the LQ.

Question 32

Define Upper Quartile (UQ).

Answer 32

The value ¾ of the way through the data; 25% of the data is above the UQ.

Question 33

How is LQ calculated for discrete data?

Answer 33

LQ = ¼(n+1)th value

Question 34

How is UQ calculated for discrete data?

Answer 34

UQ = ¾(n+1)th value

Question 35

What is the Interpercentile Range (IPR)?

Answer 35

The difference between two percentiles.

Question 36

What does a Box Plot represent?

Answer 36

Important features of the data and gives a summary of the spread/skew of the data.

Question 37

What are the five pieces of information included in a Box Plot?

Answer 37

* Minimum Value * Lower Quartile (LQ) * Median * Upper Quartile (UQ) * Maximum Value

Question 38

What is the formula for calculating standard deviation (SD) using discrete data?

Answer 38

σ = √(1/n ∑(x - x̅)²) or σ = √(∑x²/n - (∑x)²/n²)

Question 39

What does a smaller standard deviation (SD) indicate?

Answer 39

The data is closer to the mean.

Question 40

What does a larger standard deviation (SD) indicate?

Answer 40

The data is more spread out from the mean.

Question 41

What is the Interdecile Range?

Answer 41

The difference between the first and ninth deciles.

Question 42

How are outliers defined in relation to IQR?

Answer 42

Values that are more than 1.5 x IQR above UQ or below LQ.

Question 43

What is the formula to identify outliers?

Answer 43

Outliers are values > UQ + (1.5 x IQR) or < LQ - (1.5 x IQR)

Question 44

How can outliers be identified using mean and standard deviation?

Answer 44

Values more than 3 SD away from the mean.

Question 45

What does skewness describe?

Answer 45

The shape of the distribution and how the data is spread out.

Question 46

What indicates a positive skew?

Answer 46

Most values are at the beginning of the data set, with the tail going in the positive direction of the x-axis.

Question 47

What indicates a negative skew?

Answer 47

Most values are at the end of the data set, with the tail pointing towards the negative direction of the x-axis.

Question 48

What does a symmetrical distribution mean?

Answer 48

The data is evenly distributed on both sides of the median.

Question 49

What is the formula for calculating skewness?

Answer 49

Skewness = 3(means - median) / standard deviation

Question 50

When comparing data sets, what should be considered?

Answer 50

* Measure of average (mean/median/mode) * Measure of spread (range/IQR/SD) * Skewness

Question 51

What is the first step in drawing Box Plots?

Answer 51

Calculate LQ, UQ, median, and identify minimum and maximum values.

Question 52

What is the significance of the median in a Box Plot?

Answer 52

It marks the middle of the data, with 50% of the data above and below this value.

Question 53

What is the relationship between mean, median, and mode when comparing two data sets?

Answer 53

Mean/median/mode for data set A is larger than mean/median/mode for data set B, so on average, data set A is more than data set B.

Question 54

What does a larger range/IQR/SD indicate when comparing two data sets?

Answer 54

Range/IQR/SD for data set A is larger than that of data set B, so the results of data set A are more spread out/less consistant than those of data set B.

Question 55

What does a smaller range/IQR/SD imply about a data set?

Answer 55

Data set A has a smaller range/IQR/SD than data set B, which means the results for data set A are more consistant.

Question 56

How does standard deviation relate to the closeness of values to the mean?

Answer 56

Lower SD means values are closer to the mean and therefore higher SD means values are more spread out from the mean.

Question 57

What does it mean if a box plot for a data set is positively skewed?

Answer 57

Box plot for data set A is positively skewed, indicating that the majority of results were lower with few higher results.

Question 58

What does it mean if a box plot for a data set is negatively skewed?

Answer 58

Box plot for data set A is negatively skewed, indicating that the majority of results were higher with few lower results.

Question 59

What should always be referenced when interpreting data?

Answer 59

Always make reference to individual values and mention which data set is larger/smaller than the other clearly.

Question 60

When comparing averages, what terms should be used?

Answer 60

Comparing averages involves using mean, median, and mode.

Question 61

What should be included when interpreting data comparisons?

Answer 61

Always interpret in context and link back to the scenario in the question and labels on axes.

Question 62

What is the importance of pairing appropriate values when comparing data?

Answer 62

When comparing data, make sure to pair the appropriate values of average and spread.

Question 63

List the measures of average.

Answer 63

* Mode * Median * Mean

Question 64

List the measures of spread.

Answer 64

* Range * Range/IQR * Range/SD