IB Mathematics AI – Topic 4

For ungrouped data:

\[ \bar{x} = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + \cdots + x_n}{n} \]

For grouped data (frequency table):

\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]

where \(f_i\) is frequency and \(x_i\) is the data value (or midpoint for grouped intervals)

⚠️ Common Pitfalls & Tips:

• The mean is highly affected by outliers – consider using median for skewed data
• For grouped data, use the midpoint of each class interval
• Always check units and round appropriately (typically 3 significant figures for IB)

For ungrouped data:

• If \(n\) is odd: Median = \(\left(\frac{n+1}{2}\right)^{\text{th}}\) value
• If \(n\) is even: Median = \(\frac{1}{2}\left[\left(\frac{n}{2}\right)^{\text{th}} + \left(\frac{n}{2}+1\right)^{\text{th}}\right]\) value

From cumulative frequency: Median position = \(\frac{n}{2}\)

⚠️ Common Pitfalls & Tips:

• Always order the data first before finding the median
• For even-sized datasets, take the mean of the two middle values
• Use cumulative frequency curves to estimate median graphically

• Unimodal: One mode
• Bimodal: Two modes
• Multimodal: More than two modes
• No mode: All values occur with equal frequency

For grouped data: The modal class is the interval with the highest frequency

⚠️ Common Pitfalls & Tips:

• Mode is the only measure of central tendency that can be used for categorical data
• A dataset may have no mode or multiple modes
• For histograms, the modal class is the bar with the highest frequency

Solution:

(a) Mean:

\[ \bar{x} = \frac{78 + 85 + 92 + 78 + 88 + 95 + 82 + 78 + 90}{9} \]

\[ \bar{x} = \frac{766}{9} = 85.1 \text{ (3 s.f.)} \]

(b) Median:

First, order the data: 78, 78, 78, 82, 85, 88, 90, 92, 95

Since \(n = 9\) (odd), median position = \(\frac{9+1}{2} = 5^{\text{th}}\) value

Median = 85

(c) Mode:

The value 78 appears 3 times (most frequent)

Mode = 78

Solution:

Using the formula for grouped data:

\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{1505}{50} = 30.1 \text{ years} \]

The modal class is 30-39 (highest frequency = 18)

\[ \text{Range} = \text{Maximum value} - \text{Minimum value} \]

• \(Q_1\) (Lower Quartile): 25% of data lies below this value
• \(Q_2\) (Median): 50% of data lies below this value
• \(Q_3\) (Upper Quartile): 75% of data lies below this value

Interquartile Range (IQR):

\[ \text{IQR} = Q_3 - Q_1 \]

IQR measures the spread of the middle 50% of data and is resistant to outliers.

Outlier Detection:

A data point is an outlier if:

• It is less than \(Q_1 - 1.5 \times \text{IQR}\)
• It is greater than \(Q_3 + 1.5 \times \text{IQR}\)

⚠️ Common Pitfalls & Tips:

• Your GDC can calculate quartiles quickly – know how to access this function
• \(Q_2\) is always equal to the median
• IQR is preferred over range when outliers are present
• Use cumulative frequency curves to find quartiles graphically

Solution:

(a) Finding Quartiles:

Data is already ordered. \(n = 15\)

\(Q_2\) (Median) = 8th value = 25°C

\(Q_1\) = Median of lower half (first 7 values) = 4th value = 22°C

\(Q_3\) = Median of upper half (last 7 values) = 12th value = 29°C

(b) IQR:

\[ \text{IQR} = Q_3 - Q_1 = 29 - 22 = 7°\text{C} \]

(c) Outliers:

Lower boundary: \(Q_1 - 1.5 \times \text{IQR} = 22 - 1.5(7) = 11.5°\text{C}\)

Upper boundary: \(Q_3 + 1.5 \times \text{IQR} = 29 + 1.5(7) = 39.5°\text{C}\)

Since all values lie between 11.5°C and 39.5°C, there are no outliers.

Key Features:

• Bars touch (no gaps) – continuous data
• X-axis: Class intervals (bins)
• Y-axis: Frequency or frequency density
• Frequency density used when class widths are unequal

Frequency Density Formula:

\[ \text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}} \]

Area of bar = Frequency

\[ \text{Frequency} = \text{Frequency Density} \times \text{Class Width} \]

⚠️ Common Pitfalls & Tips:

• Bars must touch in histograms (unlike bar charts)
• When class widths vary, always use frequency density on y-axis
• Modal class is the bar with the highest frequency (not necessarily tallest bar if using frequency density)
• Read scales carefully – check if y-axis shows frequency or frequency density

Five-Number Summary:

1. Minimum value
2. \(Q_1\) (Lower Quartile)
3. \(Q_2\) (Median)
4. \(Q_3\) (Upper Quartile)
5. Maximum value

Interpreting Box Plots:

• Symmetrical: Median in center of box, whiskers equal length
• Positively skewed: Right whisker longer, median closer to \(Q_1\)
• Negatively skewed: Left whisker longer, median closer to \(Q_3\)

⚠️ Common Pitfalls & Tips:

• Box plots don't show individual data points or frequency
• Always plot outliers separately beyond the whiskers
• Use the scale carefully when drawing or reading box plots
• Remember: The box contains 50% of the data (middle portion)

Solution:

(a) Frequency density must be used because the class widths are not equal. The classes have widths of 20, 10, 20, and 10 minutes respectively. Using frequency density ensures that the area of each bar represents the frequency correctly.

Calculation check:

For 20-30 interval: \(\text{Frequency Density} = \frac{12}{10} = 1.2\) ✓

(b) The modal class is 30-50 minutes because it has the highest frequency (16 students), not the 20-30 class which has the tallest bar (highest frequency density).

Key Features:

• Plot cumulative frequency against the upper class boundary
• Join points with a smooth curve (ogive)
• Final point has y-coordinate equal to total frequency (\(n\))
• Used to estimate median, quartiles, and percentiles

Reading from Cumulative Frequency Curve:

• Median: Find \(\frac{n}{2}\) on y-axis, read across to curve, then down to x-axis
• \(Q_1\): Find \(\frac{n}{4}\) on y-axis
• \(Q_3\): Find \(\frac{3n}{4}\) on y-axis
• Percentiles: For \(k^{\text{th}}\) percentile, find \(\frac{kn}{100}\) on y-axis

⚠️ Common Pitfalls & Tips:

• Always plot at upper class boundaries, not midpoints
• Start cumulative frequency at 0 for the lower boundary of the first class
• Draw a smooth S-shaped curve, not straight lines between points
• Use a ruler for horizontal and vertical lines when reading values
• Check that the last cumulative frequency equals the total \(n\)

Solution:

(a) Median:

Position of median = \(\frac{n}{2} = \frac{80}{2} = 40^{\text{th}}\) value

From cumulative frequency curve: Read across from 40 on y-axis to curve, then down to x-axis

Median ≈ 163 cm

(b) Quartiles:

\(Q_1\) position = \(\frac{n}{4} = \frac{80}{4} = 20^{\text{th}}\) value → \(Q_1\) ≈ 159 cm

\(Q_3\) position = \(\frac{3n}{4} = \frac{3 \times 80}{4} = 60^{\text{th}}\) value → \(Q_3\) ≈ 167 cm

(c) IQR:

\[ \text{IQR} = Q_3 - Q_1 = 167 - 159 = 8 \text{ cm} \]

(d) Number of students between 158 cm and 167 cm:

From cumulative frequency curve:

• At 167 cm: cumulative frequency ≈ 60

• At 158 cm: cumulative frequency ≈ 16

Number of students = 60 - 16 = 44 students

Solution:

(a) IQR:

\[ \text{IQR} = Q_3 - Q_1 = 78 - 58 = 20 \]

IQR = 20 marks

(b) Outlier Test:

Lower boundary: \(Q_1 - 1.5 \times \text{IQR} = 58 - 1.5(20) = 58 - 30 = 28\)

Upper boundary: \(Q_3 + 1.5 \times \text{IQR} = 78 + 1.5(20) = 78 + 30 = 108\)

Since \(105 < 108\), a score of 105 would NOT be considered an outlier.

(c) Skewness:

Looking at the box plot structure:

• Median (67) is closer to \(Q_1\) (58) than to \(Q_3\) (78)
• Distance from \(Q_1\) to Median = 67 - 58 = 9
• Distance from Median to \(Q_3\) = 78 - 67 = 11
• Right whisker (95 - 78 = 17) is longer than left whisker (58 - 42 = 16)

The distribution is slightly positively skewed (skewed to the right) because the upper half of the data is more spread out than the lower half.

For ungrouped data (population):

\[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}} \quad \text{or} \quad \sigma = \sqrt{\frac{\sum x_i^2}{n} - \mu^2} \]

Variance: \(\sigma^2\)

For grouped data:

\[ \sigma = \sqrt{\frac{\sum f_i(x_i - \bar{x})^2}{\sum f_i}} \quad \text{or} \quad \sigma = \sqrt{\frac{\sum f_i x_i^2}{\sum f_i} - \bar{x}^2} \]

Interpretation:

• Larger standard deviation → more spread out data
• Smaller standard deviation → data clustered around mean
• Standard deviation = 0 → all values are identical

⚠️ Common Pitfalls & Tips:

• Always use your GDC for standard deviation calculations in exams
• Make sure you know the difference between \(\sigma_n\) (population) and \(\sigma_{n-1}\) (sample) on your calculator
• Standard deviation has the same units as the original data
• Variance is in squared units

Central Tendency

• Mean: Average, affected by outliers
• Median: Middle value, resistant to outliers
• Mode: Most frequent value

Dispersion

• Range: Max - Min
• IQR: \(Q_3 - Q_1\)
• Standard Deviation: Spread from mean

Visualizations

• Histograms: Frequency distribution
• Box Plots: Five-number summary
• Cumulative Frequency: Running totals

Key Formulas

• Outliers: \(Q_1 - 1.5 \times \text{IQR}\)
• Freq Density: \(\frac{\text{Freq}}{\text{Width}}\)
• Quartile positions: \(\frac{n}{4}\), \(\frac{3n}{4}\)