Chapter 2

Question 1

Definition of a Measure of Location

Answer 1

A measure of location describes the position of a data value in a data set.
Examples include the mean, median, mode, percentiles, and deciles.

Question 3

Definition of a Measure of Location

Answer 3

A measure of location describes the position of a data value in a data set.
Examples include the mean, median, mode, percentiles, and deciles.

Question 4

Mean (x̄) Formula

Answer 4

The mean is the sum of all data values divided by the number of data values.
Formula for the Mean:
x̄ = (∑x) / n

If data is in a frequency table:
x̄ = (∑fx) / (∑f)

Question 5

Median

Answer 5

The middle value when data is ordered.
For discrete data, the median is the (n + 1)/2th value.
For grouped data, estimate using interpolation

Question 6

Mode/Modal Class

Answer 6

The mode is the most frequently occurring value.
If data is grouped, the modal class is the class interval with the highest frequency.

Question 7

Choosing the Best Measure

Answer 7

Mean: Uses all data, affected by outliers.
Median: Good for skewed data, not affected by outliers.
Mode: Used for categorical data, but not always useful.

Question 8

Percentiles and Quartiles

Answer 8

Percentiles split data into 100 equal parts.
Quartiles split data into 4 equal parts:
Lower quartile (Q₁): ¼(n)th value
Median (Q₂): ½(n)th value
Upper quartile (Q₃): ¾(n)th value

Question 9

Finding Quartiles for Discrete Data

Answer 9

If n/4 is a whole number, take the midpoint of this data point and the next.
If n/4 is not a whole number, round up.

Question 10

Finding Quartiles for Grouped Data

Answer 10

Use interpolation assuming data is evenly distributed within classes.

Question 11

Interpolation formula

Answer 11

Estimated value = L + (( (n / k) - F ) / f ) × c
Where:

L = lower boundary of the class containing the percentile or quartile
n = total cumulative frequency
k = the fraction representing the desired position (e.g., for median, k = 2, for quartiles k = 4, for percentiles k = 100)
F = cumulative frequency before the class
f = frequency of the class
c = class width (upper boundary - lower boundary)

Question 12

Interquartile Range (IQR)

Answer 12

Interquartile Range (IQR) Formula:
IQR = Q3 - Q1

• Measures the spread of the middle 50% of the data.
• Not affected by extreme values (outliers).
• Helps compare variability between data sets.

Question 13

Interpercentile Range

Answer 13

The difference between two given percentiles (e.g., 10th to 90th).
Often preferred over the range since it excludes extremes.

Question 14

Variance (σ²)

Answer 14

Variance Formula (Raw Data):
σ² = (∑x² / n) - (∑x / n)²

Standard Deviation (Raw Data):
σ = √σ²

Variance (Grouped Data):
σ² = (∑fx² / ∑f) - (∑fx / ∑f)²

Standard Deviation (Grouped Data):
σ = √σ²

Question 15

Coding for Simplified Calculations

Answer 15

Coding transforms data for easier calculations:
y = (x - a) / b

New mean:
ȳ = (x̄ - a) / b

New standard deviation:
σᵧ = σₓ / b

Where:

a and b are constants.
Coding shifts and scales data, affecting the mean but not adding/subtracting constants to standard deviation.