Dotplot, Boxplot and Skewness of Nitrate Data: 18-Env-B7 Environmental Sampling and Analysis, December 2014, Q4

Question 4: Dotplot, Boxplot and Skewness of Nitrate Data (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Dotplot, Boxplot, and Fences

Given. $n=10$ nitrate values (sorted): 2, 5, 8, 9, 13, 16, 24, 45, 57, 120 (mg/L). Printed summary: Median $=14.5$, $Q_1=7.3$, $Q_3=48.0$.

Find. The dotplot and boxplot (with fences marked) and whether any outliers exist.

Approach. The standard (Tukey) boxplot fences sit $1.5\times IQR$ beyond each quartile; any data point outside a fence is flagged as an outlier and plotted individually, with the whisker itself drawn only to the most extreme non-outlier data value.

Interquartile range. $IQR=Q_3-Q_1=48.0-7.3=40.7$.
Fences. $\text{Lower fence}=Q_1-1.5\,IQR=7.3-61.05=-53.75$; $\text{Upper fence}=Q_3+1.5\,IQR=48.0+61.05=\boxed{109.05}$.
Outlier check. No data point lies below the (negative) lower fence. The maximum value, 120, exceeds the upper fence (109.05) – 120 is flagged as a single high outlier. The whisker is therefore drawn only to the largest non-outlier value, 57, not to the fence itself.

Fig. 2 – Dotplot (top) and boxplot (bottom) of the 10 nitrate values, on a shared axis. The box spans $Q_1$–$Q_3$ with the median line; whiskers extend to the most extreme non-outlier data value on each side (2 and 57); the point beyond the upper fence (120) is flagged as an outlier.

(b) Characteristics, Quartile Skew, and Coefficient of Skewness

Approach. Compare the mean to the median and use both a robust (quartile-based) and a classical (moment-based) skewness measure to characterize the shape of the distribution.

Quartile (Bowley) skew. $S_Q=\dfrac{(Q_3-\text{Median})-(\text{Median}-Q_1)}{Q_3-Q_1}=\dfrac{Q_3+Q_1-2\,\text{Median}}{Q_3-Q_1}=\dfrac{48.0+7.3-2(14.5)}{40.7}=\dfrac{26.3}{40.7}$ $$S_Q=\boxed{0.646}$$
Coefficient of skewness (Pearson). $SK=\dfrac{3(\text{Mean}-\text{Median})}{StDev}=\dfrac{3(29.9-14.5)}{36.4}=\dfrac{46.2}{36.4}$ $$SK=\boxed{1.27}$$

Both measures are positive and substantial, confirming what the boxplot already shows visually: the upper whisker (57) is far longer than the lower whisker (down to 2), the box itself is shifted toward the low end of the whisker span, and the mean (29.9) sits well above the median (14.5) – the single high value pulls the mean upward while the median, being robust to extremes, stays close to the bulk of the data. The data are therefore strongly right (positively) skewed, with one flagged high outlier (120 mg/L) driving most of that skew; a lognormal or other right-skewed model, rather than a normal distribution, would be the appropriate choice for any further statistical inference on this data set (e.g., before applying a t-test, per Question 2).

Final results – Question 4
Item	Result
IQR	40.7 mg/L
Lower / upper fence	−53.75 / 109.05 mg/L
Outlier(s)	120 mg/L (high outlier)
Whisker ends	2 mg/L (low) / 57 mg/L (high)
Quartile skew, $S_Q$	0.646 (right-skewed)
Coefficient of skewness, $SK$	1.27 (strongly right-skewed)

Concept & Theory

Topic. Exploratory data analysis (EDA): dotplots, boxplots, fences, and skewness measures for a small, outlier-contaminated data set.

Key relations. $IQR=Q_3-Q_1$; Tukey fences $=Q_{1,3}\mp1.5\,IQR$; quartile skew $S_Q=(Q_3+Q_1-2\,\text{Median})/IQR$; Pearson's coefficient of skewness $SK=3(\text{Mean}-\text{Median})/s$.

Why this works. The $1.5\times IQR$ rule is a distribution-free heuristic calibrated so that, for a normal population, only about 0.7% of points fall outside the fences by chance – making a flagged point a genuine signal worth investigating rather than routine noise. The two skewness measures are complementary: $S_Q$ is robust (uses only the quartiles/median, unaffected by the outlier's exact value) while $SK$ is sensitive to the outlier through the mean and standard deviation – their agreement in sign confirms the skew is a real feature of the bulk of the data, not an artefact of the single extreme point.

Common pitfall. Drawing the boxplot whisker all the way out to the fence itself rather than to the most extreme data value within the fence – here the fence (109.05) is not a data point at all, and the correct whisker end is 57.

Source. Walpole et al., Probability & Statistics for Engineers and Scientists, ch. 1 (descriptive statistics, boxplots); Davis & Cornwell, ch. 2 (environmental-data characteristics – right-skew, non-detects).

Question 4 of 6: Dotplot, Boxplot and Skewness of Nitrate Data

Question 4: Dotplot, Boxplot and Skewness of Nitrate Data (20 marks)

(a) Dotplot, Boxplot, and Fences

(b) Characteristics, Quartile Skew, and Coefficient of Skewness