IB Mathematics AA Understanding of box and whisker diagrams Study Notes
IB Mathematics AA Understanding of box and whisker diagrams Study Notes
IB Mathematics AA Understanding of box and whisker diagrams Notes Offer a clear explanation of Understanding of box and whisker diagrams, including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Understanding of box and whisker diagrams.
Understanding Box and Whisker Diagrams
Introduction
Box and whisker diagrams, also known as box plots, are a graphical representation of data that summarize its distribution using five key statistics: minimum, lower quartile, median, upper quartile, and maximum. This topic explores the production, interpretation, and comparison of box plots, emphasizing their application in understanding data distribution and identifying outliers.
Key Concepts
Data Presentation
- Frequency Distributions: Data can be presented in tables for both discrete and continuous variables.
- Class Intervals: These are represented as inequalities without gaps.
- Histograms: Frequency histograms are created using equal class intervals (frequency density histograms are not required).
Cumulative Frequency
- Cumulative Frequency Graphs: These are used to find key statistical measures, such as:
- Median
- Quartiles
- Percentiles
- Range and Interquartile Range (IQR)
Box and Whisker Diagrams
- Components: A box plot includes:
- Lower quartile (Q1)
- Median (Q2)
- Upper quartile (Q3)
- Minimum and maximum values (excluding outliers)
- Outliers, indicated with crosses
- Uses: Box plots are valuable for:
- Summarizing data distributions
- Identifying outliers
- Comparing multiple data sets
Interpreting Box and Whisker Diagrams
- Symmetry: Symmetric box plots may indicate normal distribution.
- Range and IQR: Analyze the spread of data using the range and interquartile range.
- Median: The median line shows the central tendency of the data.
- Comparison of Distributions: Use box plots to compare two data sets, focusing on:
- Symmetry
- Medians
- Ranges and IQRs
IB Mathematics AA SL Understanding of box and whisker diagrams Exam Style Worked Out Questions
Question
A company manufactures metal tubes for bicycle frames. The diameters of the tubes, Dmm, are normally distributed with mean 32 and standard deviation s. The interquartile range of the diameters is 0.28.
Find the value of \(\sigma \).
▶️Answer/Explanation
Answer:
METHOD 1
\(Q_1\) = 31.86 OR \(Q_3\) = 32.14 recognition that the area under the normal curve below \(Q_1\) or above \(Q_3\) is 0.25 OR the area between \(Q_1\) and \(Q_3\) is 0.5 (seen anywhere including on a diagram)
EITHER
equating an appropriate correct normal CDF function to its correct probability (0.25 or 0.5 or 0.75)
OR
z = −0.674489… OR z = 0.674489… (seen anywhere)
-0.674489… = \(\frac{31.86 – 32}{\sigma }\) OR 0.674489… = \(\frac{32.14 – 32}{\sigma }\)
THEN
0.207564…
\(\sigma = 0.208\) (mm)
METHOD 2
recognition that the area under the normal curve below \(Q_1\) or above \(Q_3\) is 0.25 OR the area between \(Q_1\) and \(Q_3\) is 0.5 (seen anywhere including on a diagram)
z = −0.674489… OR z = 0.674489…
\((Q_1=) 32 – 0.674489… \sigma \) OR \((Q_3 =) 32 + 0.674489… \sigma \)
\((Q_3 – Q_1 =) 2 \times 0.674489… \sigma \)
2 \(\times \) 0.674489… \(\sigma \) = 0.28
0.207564…
\(\sigma = 0.208\) (mm)
The interquartile range (IQR) is a measure of statistical dispersion, which is the difference between the third quartile (Q3) and the first quartile (Q1) of a dataset. For a normal distribution, the IQR is related to the standard deviation \((\sigma)\) by a factor of approximately 1.35.
\(\text{IQR} = Q_3 – Q_1 = 1.349\sigma\)
Where:
- IQR is the interquartile range
- Q3 is the 75th percentile
- Q1 is the 25th percentile
- σ is the standard deviation
The interquartile range of the diameters is 0.28.
We can substitute the known IQR value into the equation: \(0.28 = 1.349\sigma \)
\(\sigma = \frac{0.28}{1.349} \)
\(\sigma \approx 0.208\)
Question
A survey at a swimming pool is given to one adult in each family. The age of the adult, a years old, and of their eldest child, c years old, are recorded.
The ages of the eldest child are summarized in the following box and whisker diagram.
diagram not to scale
(a) Find the largest value of c that would not be considered an outlier.
The regression line of a on c is \(a = \frac{7}{4}c + 20.\) The regression line of c on a is \(c = \frac{1}{2}a – 9.\)
(b) (i) One of the adults surveyed is 42 years old. Estimate the age of their eldest child.
(ii) Find the mean age of all the adults surveyed.
▶️Answer/Explanation
Ans:
(a) IQR = 10 – 6 (=4)
attempt to find Q3 + 1.5 × IQR
10 + 6
16
(b)
(i) choosing c = \(\frac{1}{2}a – 9\)
\(\frac{1}{2}\times 42-9\)
= 12 (years old)
(ii) attempt to solve system by substitution or elimination
34 (years old)