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.
Presentation of Data: Frequency Distributions
Presentation of Data: Frequency Distributions
Frequency distributions organize data to show how often each value or range of values occurs in a dataset.
Discrete Data
Discrete data take specific values, often whole numbers (e.g., number of children).
| Number of Children | Frequency |
|---|---|
| 0 | 3 |
| 1 | 5 |
| 2 | 7 |
| 3 | 4 |
Continuous Data
Continuous data can take any value within a range (e.g., height, weight).
| Height (cm) | Frequency |
|---|---|
| 150 – 159 | 2 |
| 160 – 169 | 6 |
| 170 – 179 | 9 |
| 180 – 189 | 3 |
Notes:
- Discrete frequency tables list each value individually with its frequency.
- Continuous frequency tables group data into intervals (class intervals).
- Frequency tables make it easier to construct histograms, bar charts, and cumulative frequency graphs.
Example
The following are test scores of 20 students:
45, 56, 67, 67, 45, 89, 90, 56, 56, 67, 78, 45, 56, 67, 90, 100, 89, 45, 56, 90
▶️ Answer/Explanation
| Score | Frequency |
|---|---|
| 45 | 4 |
| 56 | 5 |
| 67 | 4 |
| 78 | 1 |
| 89 | 2 |
| 90 | 3 |
| 100 | 1 |
Frequency Histograms with Equal Class Intervals
Frequency Histograms with Equal Class Intervals
A frequency histogram displays data using bars to represent the frequency of observations in each class interval.
- For equal class intervals, all bars will have the same width. The height of each bar corresponds to the frequency.
- The x-axis represents the class intervals (bins) and the y-axis represents frequency.
- Histograms are useful for visualizing the distribution of continuous data.
Example
The test scores of 20 students were grouped into equal-width intervals:
| Score Interval | Frequency |
|---|---|
| 40 – 49 | 4 |
| 50 – 59 | 5 |
| 60 – 69 | 4 |
| 70 – 79 | 1 |
| 80 – 89 | 2 |
| 90 – 99 | 3 |
| 100 – 109 | 1 |
▶️ Answer/Explanation
The histogram would have:
- Bars of equal width (interval = 10 units).
- Heights of bars proportional to frequency (e.g., tallest bar at 5 for 50-59 interval).
- The x-axis labeled with score intervals, y-axis with frequency.
Example: The following table shows the distribution of test scores:
▶️ Answer/Explanation
|
Cumulative Frequency & Graphs
Cumulative Frequency & Graphs
Cumulative frequency is the running total of frequencies up to and including a given class boundary.
A cumulative frequency graph (or ogive) is a plot of cumulative frequency against the upper class boundary.
The graph can be used to estimate key statistics: median, quartiles (Q1, Q3), percentiles, range, and interquartile range (IQR).
- Median: The value at the 50th percentile (middle cumulative frequency).
- Lower quartile (Q1): The value at the 25th percentile.
- Upper quartile (Q3): The value at the 75th percentile.
- Interquartile range (IQR): Q3 – Q1; measures spread of middle 50% of data.
Example
The heights (in cm) of 20 students are:
145, 150, 152, 153, 155, 158, 160, 162, 163, 165, 166, 168, 170, 172, 173, 175, 178, 180, 182, 185
Group the data into class intervals of 10 cm and create a cumulative frequency table.
▶️ Answer/Explanation
| Height (cm) | Frequency | Cumulative Frequency |
|---|---|---|
| 140 – 149 | 1 | 1 |
| 150 – 159 | 4 | 5 |
| 160 – 169 | 5 | 10 |
| 170 – 179 | 5 | 15 |
| 180 – 189 | 5 | 20 |
The cumulative frequency shows how many students have heights up to the upper limit of each class interval.
Example
The following table shows the test scores of 40 students:
| Score (≤) | Cumulative Frequency |
|---|---|
| 40 | 3 |
| 50 | 9 |
| 60 | 17 |
| 70 | 26 |
| 80 | 33 |
| 90 | 37 |
| 100 | 40 |
▶️ Answer/Explanation
Median: The 20th value (since $40 ÷ 2 = 20).$
Locate 20 on the cumulative frequency graph → estimate score $≈ 62$
Lower quartile (Q1): 10th value $(40 × 0.25 = 10)$
Estimate score $≈ 52$
Upper quartile (Q3): 30th value $(40 × 0.75 = 30)$
Estimate score $≈ 75$
IQR: $Q_3 – Q_1 = 75 – 52 = 23$
A cumulative frequency graph would plot the points:
- (40, 3), (50, 9), (60, 17), (70, 26), (80, 33), (90, 37), (100, 40)
Box and Whisker Diagrams
Box and Whisker Diagrams
A box and whisker diagram (or box plot) is a graphical representation of data that shows the distribution’s central value, spread, and possible outliers. It is useful for comparing two or more data sets.
Key features of a box plot:
- Minimum value: the smallest data point (excluding outliers).
- Lower quartile (Q1): 25% of the data lies below this value.
- Median (Q2): the middle value that divides the data into two halves.
- Upper quartile (Q3): 75% of the data lies below this value.
- Maximum value: the largest data point (excluding outliers).
- Outliers: values that lie more than 1.5 IQR below Q1 or above Q3, often marked with a cross (×).
How to construct a box and whisker diagram:
- Order the data set from smallest to largest.
- Find the median, Q1, and Q3.
- Calculate the interquartile range (IQR = Q3 − Q1).
- Determine and mark any outliers (points beyond 1.5 × IQR from the quartiles).
- Draw a box from Q1 to Q3, mark the median inside the box.
- Extend “whiskers” from the box to the minimum and maximum values (excluding outliers).
How to compare two box plots:
- Symmetry: Check if the median is centered in the box and if whiskers are of equal length → indicates symmetry.
- Spread: Compare IQR and overall range to see which data set is more variable.
- Median: See which data set tends to have higher or lower values.
- Outliers: Identify data sets with unusual extreme values.
Link to Normal Distribution: If the box is roughly symmetric with no outliers and whiskers of similar length, the data may be normally distributed.
Example:
Draw Box Plot Diagra for The following data are the heights of 40 students in a statistics class:
$59, 60, 61, 62, 62, 63, 63, 64, 64, 65, 65, 65, 65, 65, 65, 66, 66, 66, 67, 68, 68, 69, 70, 70, 70, 70, 70, 71, 71, 72, 72, 73, 74, 74, 75, 77$
▶️ Answer/Explanation
- Minimum value: 59
- Maximum value: 77
- Q1 (First quartile): 64.5
- Q2 (Median): 66
- Q3 (Third quartile): 70
Key calculations:
- Range = 77 – 59 = 18
- IQR = Q3 – Q1 = 70 – 64.5 = 5.5
Interpretation:
- Each quarter contains about 25% of the data.
- 1st quarter spread = 64.5 – 59 = 5.5
- 2nd quarter spread = 66 – 64.5 = 1.5
- 3rd quarter spread = 70 – 66 = 4
- 4th quarter spread = 77 – 70 = 7
- The second quarter has the smallest spread, the fourth has the largest.
- The interval 59–65 has more than 25% of the data.
Box Plot Diagram:
Example:
Construct a Box and Whisker Plot using TI-83/84
The following data are the ages of participants in a survey:
$23, 25, 30, 22, 28, 24, 26, 29, 30, 27, 25, 28, 31, 24, 22$
▶️ Answer/Explanation
- Press
STAT→ select 1:Edit. - Enter the data into
L1. Input: 23, 25, 30, 22, 28, 24, 26, 29, 30, 27, 25, 28, 31, 24, 22. - Press
2nd→Y=(STAT PLOT). - Choose 1:Plot1 → turn it ON.
- Select the box plot icon (with or without outliers, as required).
- Press
ZOOM→ choose 9:ZoomStat to automatically fit the data on the screen.
The calculator will display:
- Minimum = 22
- Q1 = 24
- Median (Q2) = 26
- Q3 = 29
- Maximum = 31
This box plot helps visualize the spread and symmetry of the age data.