Measures of Position

Position measures tell you where a value ranks within a dataset. If you scored at the 80th percentile, you did better than 80% of everyone. Quartiles divide sorted data into four equal chunks: Q1 (25th percentile), Q2 = median (50th), Q3 (75th). The IQR = Q3 − Q1 captures the middle 50% of data and is used to flag outliers — any value below Q1 − 1.5·IQR or above Q3 + 1.5·IQR is considered a potential outlier. A z-score tells how many standard deviations a value is from the mean: z = 0 means exactly at the mean, z = 2 means two standard deviations above it.

$$z=\frac{x-\mu}{\sigma}, \quad IQR=Q_3-Q_1$$

Quartile Position

For the ordered data 5, 8, 9, 12, 15, 18, 20, 24, 30, find $Q_1$ and $Q_3$ using the median-of-halves method.

Lower half: 5, 8, 9, 12. Thus $Q_1=(8+9)/2=8.5$.

Upper half: 18, 20, 24, 30. Thus $Q_3=(20+24)/2=22$.

Z-Score Interpretation

A test has mean 72 and standard deviation 8. A student scores 88. Find the z-score.

$$z=\frac{88-72}{8}=2.00$$

The score is 2 standard deviations above the mean.

Percentile Rank

In a class of 50 students, 38 scored lower than Ana. Estimate Ana's percentile rank.

$$PR=\frac{38}{50}(100)=76$$

Final answer: Ana is approximately at the 76th percentile.

Interquartile Range (IQR)

Find the IQR of: 4, 7, 8, 10, 13, 15, 18, 20.

Split at the median. Lower half: 4, 7, 8, 10 → Q1 = (7+8)/2 = 7.5. Upper half: 13, 15, 18, 20 → Q3 = (15+18)/2 = 16.5.

$$IQR=Q_3-Q_1=16.5-7.5=9$$

Final answer: IQR = 9.

Outlier Test Using IQR

For the data set 5, 6, 7, 8, 9, 10, 35, determine whether 35 is an outlier using the 1.5 × IQR rule. Q1 = 6, Q3 = 10.

IQR = 10 − 6 = 4. Compute fences:

$$\text{Lower fence}=Q_1-1.5(IQR)=6-6=0$$

$$\text{Upper fence}=Q_3+1.5(IQR)=10+6=16$$

Since 35 > 16, the value 35 is an outlier.

Decile Calculation

From a sorted data set of 30 values, find the position of the 4th decile (D4).

Deciles divide data into 10 equal parts. The $k$th decile position is at $\frac{k(n+1)}{10}$.

$$D_4 \text{ position}=\frac{4(31)}{10}=12.4$$

D4 is between the 12th and 13th values. Interpolate: $x_{12}+0.4(x_{13}-x_{12})$. (Substitute actual values once known.)

Finding x from a Z-Score

Exam scores are normally distributed with mean 75 and standard deviation 8. A student has a z-score of −1.5. What was the student's actual score?

Rearrange the z-score formula: $x = \mu + z\sigma$.

$$x=75+(-1.5)(8)=75-12=63$$

Final answer: The student scored 63.

Five-Number Summary

For the data 3, 7, 8, 10, 12, 15, 17, 20, 22, construct the five-number summary.

The five numbers are: Min, Q1, Median, Q3, Max.

Min = 3, Max = 22. With 9 values, median = 5th value = 12. Lower half: 3,7,8,10 → Q1 = 7.5. Upper half: 15,17,20,22 → Q3 = 18.5.

$$\text{Summary: } 3,\ 7.5,\ 12,\ 18.5,\ 22$$