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.
Show Solution 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.
Show Solution $$z=\frac{88-72}{8}=2.00$$
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.
Show Solution $$PR=\frac{38}{50}(100)=76$$
Final answer: Ana is approximately at the 76th percentile.
Interquartile Range (IQR) Find the IQR of: 4, 7, 8, 10, 13, 15, 18, 20.
Show Solution 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.
Show Solution 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).
Show Solution 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?
Show Solution 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.
Show Solution 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.
$$\text{Summary: } 3,\ 7.5,\ 12,\ 18.5,\ 22$$
Exam Generator Problems
Additional board-style practice items for this topic.
Question Bank: t17
MSTE - Statistics and Probability / Statistics / Civil Engineering Refresher
A survey on the number of pets per student found a class average of 2. For student counts of 4, 6, 10, 0, K, and 2 corresponding to 0-5 pets, find K.
4
2
3
5
Show Solution
Mean $= \frac{\sum(\text{pets}\times\text{students})}{\sum\text{students}} = 2$.
