Finding the Range

Find the range of these test scores: $72, 85, 91, 68, 77, 83, 95$

Identify the maximum value: Looking at all values: $72, 85, 91, 68, 77, 83, 95$ = Maximum $= 95$

Identify the minimum value: The smallest score in the set = Minimum $= 68$

Calculate the range: $\text{Range} = 95 - 68$ = $\text{Range} = 27$

Answer: The range is $27$ points, meaning the scores span 27 points from lowest to highest.

Finding the IQR

Find the IQR of: $12, 15, 18, 22, 25, 28, 31, 35, 42$

Order the data and find the median: Data is already ordered. With 9 values, the median is the 5th value. = Median $= 25$

Find Q1 (median of lower half): Lower half: $12, 15, 18, 22$. Median of 4 values = average of middle two: $(15 + 18) \div 2$ = $Q_1 = 16.5$

Find Q3 (median of upper half): Upper half: $28, 31, 35, 42$. Median = $(31 + 35) \div 2$ = $Q_3 = 33$

Calculate IQR: $\text{IQR} = Q_3 - Q_1 = 33 - 16.5$ = $\text{IQR} = 16.5$

Answer: The IQR is $16.5$, meaning the middle 50% of the data spans 16.5 units.

Comparing Range and IQR with Outliers

Data set: $20, 22, 24, 25, 26, 28, 30, 100$. Find both range and IQR. Which measure is more useful here?

Calculate the range: $\text{Range} = 100 - 20$ = $\text{Range} = 80$

Find the median: 8 values: median = average of 4th and 5th values = $(25 + 26) \div 2$ = Median $= 25.5$

Find Q1: Lower half: $20, 22, 24, 25$. Median = $(22 + 24) \div 2$ = $Q_1 = 23$

Find Q3: Upper half: $26, 28, 30, 100$. Median = $(28 + 30) \div 2$ = $Q_3 = 29$

Calculate IQR: $\text{IQR} = 29 - 23$ = $\text{IQR} = 6$

Compare and interpret: Range ($80$) is heavily influenced by the outlier $100$. IQR ($6$) shows the typical spread. = IQR is more useful

Answer: Range $= 80$, IQR $= 6$. The IQR is more useful because the range is inflated by the outlier (100). The typical data spread is only 6 units.

Mistake: Not ordering the data before finding quartiles

Why: Quartiles are based on position in ordered data. Unordered data gives wrong Q1 and Q3 values.

Correct: Always sort the data from smallest to largest first, then find the median and quartiles.

Mistake: Including the median in both halves when finding Q1 and Q3

Why: When you have an odd number of data points, the median should be excluded from both halves.

Correct: For odd-sized datasets, split the data excluding the median, then find Q1 and Q3 from those halves.

Mistake: Confusing range with IQR

Why: They measure different things: range uses extreme values, IQR uses quartiles.

Correct: Range $=$ Max $-$ Min. IQR $= Q_3 - Q_1$. Both measure spread, but IQR focuses on the middle 50%.

Salary Comparisons

Companies use IQR to understand typical salary ranges without being skewed by executive pay.

If salaries at a company range from 30,000 to 500,000 euros, but the IQR is 35,000 to 55,000 euros, most employees earn in that narrower range.

Athletic Performance

Coaches analyze athlete consistency using IQR of performance metrics.

A sprinter's 100m times: $11.2, 11.4, 11.3, 11.5, 11.4, 11.6, 11.3$ seconds. A small IQR indicates consistent performance.

Q: Why is IQR better than range for some datasets?

A: IQR ignores extreme values (outliers) that can inflate the range. It focuses on where the middle 50% of data lies, giving a better picture of typical spread.

Q: What does it mean if the IQR is 0?

A: An IQR of 0 means the middle 50% of your data are all the same value. This indicates very little variability in the central portion of your dataset.

Q: How do I know if I should include the median when finding Q1 and Q3?

A: With an odd number of data points, exclude the median from both halves. With an even number, split the data exactly in half and include all values.