# Data presentation and summary statistics

> IB Mathematics AA HL · IB AA HL 2021+
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u4-data-presentation-and-summary-statistics/

This sub-topic covers organizing, displaying, and summarizing univariate data for IB AA HL. You will learn to create appropriate graphical displays and calculate numerical measures to describe the center, spread, and shape of data distributions.

**Prerequisites:** [Basic understanding of categorical and quantitative data types](https://www.owlsprep.com/study/ib-math-aa-hl-u4-introduction-to-statistics/)

## Learning objectives

- Construct appropriate graphical displays for different data types
- Calculate and interpret key summary statistics for univariate data
- Identify outliers and assess their impact on summary measures
- Compare distributions using summary statistics and graphical displays

## Graphical Displays for Different Data Types

**Univariate data** — Data that records a single characteristic from each sampling unit, used for descriptive analysis

*Example:* Heights of 50 12th grade students is univariate; height + weight is bivariate

Different data types require different displays to clearly show patterns. Categorical data uses bar charts or pie charts, discrete quantitative data uses bar charts, and continuous data uses histograms, frequency polygons, or cumulative frequency curves.

> **tip**
>
> Always label your axes with units when drawing graphs for exams. Missing labels cost easy, avoidable marks.

**Worked example:** A teacher records absences per student in a 30-person class: | Absences | 0 | 1 | 2 | 3 | 4 | | Frequency | 12 | 10 | 5 | 2 | 1 | What display is most appropriate, and how should it be structured?

1. 1. Identify data type: Number of absences is discrete quantitative data.
2. 2. Match to display: Discrete data uses a separated bar chart (not a histogram, which implies continuity).
3. 3. Axes setup: X-axis labeled "Number of absences" with ticks at 0–4, Y-axis labeled "Frequency" ranging 0–12.
4. 4. Draw separated bars with height equal to the frequency for each absence value.

*Calculator:* allowed

## Measures of Center and Spread

**Interquartile Range (IQR)** — The spread of the middle 50% of ordered data, a measure of spread that is robust to outliers

*Notation:* IQR = Q_3 - Q_1

*Example:* For [2, 3, 5, 7, 11], Q1=3, Q3=7, IQR = 4

Measures of center describe the typical value of a distribution, while measures of spread describe how much data varies. Common pairs used are mean and standard deviation, or median and IQR.

- Mean = average of all data, affected by extreme outliers
- Median = middle value of ordered data, not affected by outliers
- Standard deviation = average distance of points from the mean, same units as original data

**Worked example:** Find the mean, median, and population standard deviation for [12, 15, 18, 22, 25], where this is the full population.

1. Calculate the mean:
2. $$\bar{x} = \frac{12 + 15 + 18 + 22 + 25}{5} = 18.4$$
3. Find the median: ordered data, the 3rd value is the median: Median = 18
4. Calculate sum of squared deviations from the mean:
5. $$(12-18.4)^2 + ... + (25-18.4)^2 = 109.2$$
6. Population standard deviation is the square root of average squared deviation:
7. $$\sigma = \sqrt{\frac{109.2}{5}} \approx 4.67$$

*Calculator:* allowed

## Outliers and Box Plots

**Outlier (1.5 × IQR rule)** — An observation is an outlier if it is less than $Q_1 - 1.5 \times IQR$ or greater than $Q_3 + 1.5 \times IQR$

*Example:* Q1=10, Q3=20, IQR=10: any value < -5 or > 35 is an outlier

Box plots (box-and-whisker plots) display the five-number summary (minimum, Q1, median, Q3, maximum) and explicitly mark outliers. They are ideal for comparing two or more distributions side-by-side.

**Worked example:** Identify any outliers in [4, 6, 7, 8, 10, 12, 35] using the 1.5×IQR rule.

1. Data is ordered, n=7. Median = 4th term = 8. Split into lower half [4,6,7] and upper half [10,12,35], excluding the median.
2. Q1 = median of lower half = 6, Q3 = median of upper half = 12
3. Calculate outlier boundaries:
4. $$IQR = 12 - 6 = 6 \\ Lower = 6 - 1.5(6) = -3 \\ Upper = 12 + 1.5(6) = 21$$
5. 35 is greater than 21, so 35 is classified as an outlier.

*Calculator:* allowed

## Cumulative Frequency Graphs

A cumulative frequency graph plots cumulative frequency (total observations up to a given value) against the upper bound of class intervals. It is used to estimate medians, quartiles, and percentiles from grouped continuous data.

**Worked example:** For 100 plant heights, cumulative frequency is: 10 at 10cm, 40 at 20cm, 75 at 30cm, 100 at 40cm. Estimate the median and IQR.

1. Median is the height at cumulative frequency $\frac{100}{2} = 50$.
2. Interpolate between (20cm, 40) and (30cm, 75): $20 + \frac{50-40}{75-40} \times 10 \approx 22.9$ cm
3. Q1 is at cumulative frequency 25, Q3 at 75:
4. Q1 $\approx 10 + \frac{25-10}{40-10} \times 10 = 15$ cm, Q3 = 30 cm (given)
5. Calculate IQR:
6. $$IQR = 30 - 15 = 15 \text{ cm}$$

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using a histogram for discrete categorical data
  - Why it fails: Connected histogram bars imply the variable is continuous, which is misleading and loses marks
  - Correct: Use separated bar charts for discrete or categorical data
- **Wrong:** Using the wrong standard deviation (sample vs population) from a GDC
  - Why it fails: IB expects you to select the correct value for the context, using the wrong one gives an incorrect answer
  - Correct: Use $\sigma$ for full population data, $s$ for sample data from a larger population
- **Wrong:** Using mean instead of median for skewed distributions with outliers
  - Why it fails: The mean is pulled towards the tail of skewed distributions and extreme outliers, so it does not reflect the typical value
  - Correct: Report median as the measure of center for skewed distributions or datasets with outliers
- **Wrong:** Including the overall median when calculating Q1/Q3 for odd n
  - Why it fails: This leads to incorrect quartile values and wrong outlier classification
  - Correct: Exclude the overall median from both the lower and upper halves when n is odd before calculating Q1/Q3

## Cheatsheet

| Measure Type | Name | Key Rule | Robust to Outliers? |
| --- | --- | --- | --- |
| Center | Mean | $\bar{x} = \frac{\sum x_i}{n}$ | No |
| Center | Median | Middle value of ordered data | Yes |
| Spread | Range | Max - Min | No |
| Spread | IQR | Q3 - Q1 | Yes |
| Spread | Population SD | $\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$ | No |
| Spread | Sample SD | $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$ | No |
| Outlier Rule | 1.5×IQR | Outlier if < Q1 - 1.5IQR or > Q3 + 1.5IQR | N/A |

## What's next

Data presentation and summary statistics is the foundation for all statistical work in IB AA HL. These descriptive tools help you explore and summarize patterns in raw data, and you will use summary measures throughout all remaining statistics topics. You will rely on the properties of mean and standard deviation when working with probability distributions, use box plots to compare groups, and interpret summary statistics for regression analysis. Note that formal inferential procedures such as hypothesis testing belong to Math AI, not AA, and are not part of AA HL.

- [Normal Distribution](https://www.owlsprep.com/study/ib-math-aa-hl-u4-normal-distribution/)
- [Basic probability concepts and rules](https://www.owlsprep.com/study/ib-math-aa-hl-u4-basic-probability-concepts-and-rules/)

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