Welcome to Data Presentation and Interpretation
Welcome to one of the most practical and scoring topics in your CCEA AS 2: Applied Mathematics (Statistics) module! In our data-driven world, huge collections of numbers are generated every second. But raw numbers on their own are messy and hard to read. In this chapter, you will master how to clean data, present it visually using graphs and diagrams, summarize it with key numerical measures, and find trends between paired variables.
Don't worry if statistics has ever felt abstract or confusing before. We will break down every single formula and method into clear, step-by-step pieces with plenty of everyday examples.
---1. Classifying Data and Data Cleaning
Types of Data
Before doing any calculations, you must recognize what kind of data you are handling:
• Categorical (Qualitative) Data: Non-numerical labels or descriptive categories (e.g. eye colour, car brand, gender).
• Numerical (Quantitative) Data: Data made up of numbers and quantities. This splits into two crucial types:
– Discrete Data: Values that can only take specific, separate numbers—usually counts (e.g. number of goals scored in a match: \(0, 1, 2, 3\), shoe sizes like \(7, 7.5, 8\)). You cannot score \(2.34\) goals!
– Continuous Data: Measurements that can take any real value within a given range, depending on how accurately you measure (e.g. running times: \(12.4\text{ s}\), mass: \(65.38\text{ kg}\), height: \(1.75\text{ m}\)).
• Univariate vs Bivariate Data:
– Univariate data involves just one single variable (e.g. recording the heights of 30 students).
– Bivariate data involves pairs of measurements for two variables, written as \((x, y)\) (e.g. recording both revision hours \(x\) and test score \(y\) for each student).
Data Cleaning
Real-world data is rarely perfect. Data cleaning is the process of inspecting raw data to identify, correct, or remove errors, impossible values, and obvious outliers before any calculations are performed (e.g. fixing a typo where a human height was entered as \(175\text{ m}\) instead of \(1.75\text{ m}\)).
Key Takeaway: Always ask yourself: "Is this counted (discrete) or measured on a scale (continuous)?" and "Am I tracking one variable (univariate) or two related variables (bivariate)?"
---2. Visualizing Data: Statistical Diagrams
Histograms (For Continuous Grouped Data)
A histogram looks similar to a bar chart, but there is a major mathematical difference: on a histogram, frequency is represented by the AREA of the bar, not simply its height!
The fundamental Area Principle states:
\(\text{Area of bar} = k \times \text{Frequency}\)
In standard CCEA exam questions where the vertical scale is directly calibrated (\(k = 1\)):
\(\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}\)
\(\text{Frequency} = \text{Frequency Density} \times \text{Class Width}\)
Crucial Rule for Class Boundaries: Always check for gaps between grouped intervals. For example, if discrete ages are grouped as \(10 - 19\) and \(20 - 29\), the true continuous boundaries are \(9.5 \le \text{age} < 19.5\) (class width \(= 19.5 - 9.5 = 10\)), not \(19 - 10 = 9\).
Stem-and-Leaf Diagrams
Stem-and-leaf diagrams order raw numerical data while keeping the original values visible:
• Leaves must always be single digits, placed in numerical order, and spaced evenly in columns.
• You must always include a Key showing place value (e.g. \(1 \mid 4 = 14\text{ marks}\) or \(2 \mid 3 = 2.3\text{ kg}\)). Without a key, marks are lost!
Box-and-Whisker Plots (Box Plots)
A box plot provides a visual summary of the distribution using the 5-figure summary:
1. Minimum value (lowest non-outlier)
2. Lower Quartile (\(Q_1\)) – the \(25\text{th}\) percentile
3. Median (\(Q_2\)) – the \(50\text{th}\) percentile (the vertical line inside the box)
4. Upper Quartile (\(Q_3\)) – the \(75\text{th}\) percentile
5. Maximum value (highest non-outlier)
Any extreme outliers are plotted as individual crosses (\(\times\)) or dots beyond the whiskers.
Cumulative Frequency Diagrams
Cumulative frequency is a running total of frequencies. When plotting:
• Always plot cumulative frequency points at the upper class boundary of each class interval.
• Join the points smoothly with an S-shaped curve (or polygon) starting from the lower boundary of the first interval at a cumulative frequency of \(0\).
Key Takeaway: In histograms, height \(= \text{frequency density}\) and area \(= \text{frequency}\). For cumulative frequency graphs, always plot at the upper boundary.
---3. Measures of Central Tendency (Location)
These measures tell us where the "centre" or typical balance point of our data lies.
The Mean (\(\bar{x}\))
• From raw data: Add all values and divide by the total number of items \(n\):
\(\bar{x} = \frac{\sum x}{n}\)
• From a frequency table / grouped continuous data:
\(\bar{x} = \frac{\sum fx}{\sum f} = \frac{\sum fx}{n}\)
(For grouped data, \(x\) represents the midpoint of each class interval.)
The Median (\(Q_2\)) and Linear Interpolation
The median is the middle observation that splits ordered data in half. For continuous data presented in a grouped frequency table, we estimate the median using Linear Interpolation.
The Linear Interpolation Formula:
\(Q_2 = L + \left(\frac{\frac{n}{2} - F}{f}\right) \times w\)
Where:
• \(L\) = true lower class boundary of the median class
• \(n\) = total frequency (\(\sum f\))
• \(F\) = cumulative frequency up to the start of the median class
• \(f\) = frequency of the median class
• \(w\) = class width of the median class
Example: To find the Lower Quartile (\(Q_1\)), replace \(\frac{n}{2}\) with \(\frac{n}{4}\). To find the Upper Quartile (\(Q_3\)), replace \(\frac{n}{2}\) with \(\frac{3n}{4}\).
The Mode and Modal Class
• Mode: The single most frequently occurring value in a dataset.
• Modal Class: The class interval that has the highest frequency density (or highest frequency when class widths are equal).
Key Takeaway: The mean uses every value (sensitive to extreme values), the median is resistant to extreme outliers, and the mode represents the most common observation.
---4. Measures of Spread (Variation)
Spread tells us how clustered or scattered the data points are around the centre.
Range and Interquartile Range (IQR)
• Range: \(\text{Range} = \text{Maximum} - \text{Minimum}\) (highly sensitive to extreme values).
• Interquartile Range (IQR): Measures the spread of the middle \(50\%\) of the data:
\(\text{IQR} = Q_3 - Q_1\)
• Interpercentile / Interdecile Range: Measures spread between chosen percentiles, such as the \(10\text{th}-90\text{th}\) interpercentile range (\(P_{90} - P_{10}\)).
Variance (\(\sigma^2\)) and Standard Deviation (\(\sigma\))
Standard deviation measures the average distance of observations from the mean.
• Variance for raw data:
\(\sigma^2 = \frac{\sum (x - \bar{x})^2}{n} = \frac{\sum x^2}{n} - \bar{x}^2\)
Memory Trick: "Mean of the squares minus the square of the mean."
• Variance for a frequency distribution:
\(\sigma^2 = \frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2\)
• Standard Deviation (\(\sigma\)): The positive square root of variance:
\(\sigma = \sqrt{\sigma^2} = \sqrt{\frac{\sum fx^2}{\sum f} - \bar{x}^2}\)
Key Takeaway: Standard deviation is in the original units of measurement (e.g. \(\text{cm}\)), whereas variance is in squared units (e.g. \(\text{cm}^2\)).
---5. Detecting Outliers
An outlier is an extreme value that does not fit the overall pattern of the data. In your exam, you will be given a specific rule to test for outliers.
Method 1: The Quartile / IQR Rule
A value \(x\) is an outlier if:
• \(x < Q_1 - 1.5 \times \text{IQR}\) (Lower Outlier Boundary)
• \(x > Q_3 + 1.5 \times \text{IQR}\) (Upper Outlier Boundary)
Method 2: The Mean and Standard Deviation Rule
Depending on the question instructions, a value \(x\) is considered an outlier if it lies more than 2 (or 3) standard deviations away from the mean:
• Outside the interval \(\bar{x} \pm 2\sigma\) (or \(\bar{x} \pm 3\sigma\))
Key Takeaway: Never guess an outlier by eye. Calculate the upper and lower boundary thresholds, state them clearly, and check if any values fall outside.
---6. Linear Transformations (Coding Data)
Coding simplifies large or awkward numbers into smaller, manageable numbers. Suppose we code a variable \(x\) using a linear transformation:
\(y = \frac{x - a}{b}\) \(\iff\) \(x = by + a\)
Here is how coding affects summary statistics:
1. Measures of Location (Mean, Median, Mode, Quartiles)
Location measures are affected by both addition/subtraction and multiplication/division:
\(\bar{x} = b\bar{y} + a\)
2. Measures of Spread (Standard Deviation, IQR, Range)
Spread is only affected by multiplying or dividing by the scale factor \(b\). Adding or subtracting \(a\) merely shifts all values along the number line without changing how spread out they are!
\(\sigma_x = |b| \times \sigma_y\)
\(\text{IQR}_x = |b| \times \text{IQR}_y\)
\(\sigma_x^2 = b^2 \times \sigma_y^2\)
Quick Review: If daily temperatures \(x\) in \({}^\circ\text{C}\) are converted using \(y = x - 20\), the mean decreases by \(20\), but the standard deviation \(\sigma\) remains completely unchanged!
---7. Bivariate Data: Scatter Diagrams, Correlation & Regression
Scatter Diagrams
A scatter diagram plots paired bivariate data \((x, y)\). The explanatory (independent) variable is always placed on the horizontal \(x\)-axis, and the response (dependent) variable on the vertical \(y\)-axis.
Product Moment Correlation Coefficient (PMCC / \(r\))
The PMCC, denoted by \(r\), measures the strength and direction of a linear relationship between two continuous variables:
\(-1 \le r \le +1\)
• \(r = +1\): Perfect positive linear correlation (as \(x\) increases, \(y\) increases in a straight line).
• \(r \approx 0\): No linear correlation (points are randomly scattered, or the relationship is non-linear).
• \(r = -1\): Perfect negative linear correlation (as \(x\) increases, \(y\) decreases in a straight line).
Correlation vs Causation
Did you know? High sales of ice cream correlate strongly with high numbers of shark attacks! Does eating ice cream attract sharks? No—a third confounding variable (hot summer weather) causes both.
Exam Golden Rule: Correlation does NOT imply causation! A strong correlation simply shows an association; it does not prove that one variable causes the change in the other.
The Least Squares Regression Line
The equation of the regression line of \(y\) on \(x\) is written as:
\(y = a + bx\)
• \(b\) is the gradient (the change in \(y\) for each 1-unit increase in \(x\)).
• \(a\) is the \(y\)-intercept (the value of \(y\) when \(x = 0\)).
• The regression line always passes through the point of means: \((\bar{x}, \bar{y})\).
Making Predictions: Interpolation vs Extrapolation
• Interpolation: Estimating a value of \(y\) for an \(x\)-value within the range of the original data. This is reliable.
• Extrapolation: Estimating a value of \(y\) for an \(x\)-value outside the range of observed data. This is unreliable because the linear trend may not continue beyond the observed range.
Key Takeaway: Use \(y = a + bx\) only to predict \(y\) from \(x\) within the range of given data.
---8. Common Examiner Pitfalls to Avoid
Pitfall 1: Histograms with Unequal Class Widths
Never draw bar heights equal to frequencies! Always calculate \(\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}\) and plot that on the vertical axis.
Pitfall 2: Forgetting Boundary Adjustments in Grouped Data
For continuous linear interpolation or histograms, ensure you use the true lower boundary \(L\) (e.g. for an integer interval \(20 - 24\), \(L = 19.5\), not \(20\)).
Pitfall 3: Coding Spread Errors
When decoding standard deviation from coded variable \(y = \frac{x - 50}{10}\), do not subtract \(50\)! Only the scale factor applies: \(\sigma_x = 10 \times \sigma_y\).
Pitfall 4: Misinterpreting \(r = 0\)
An \(r\) value close to \(0\) means there is no linear relationship. A strong quadratic or circular relationship could still exist between the variables.
Pitfall 5: Unjustified Extrapolation
If an exam question asks whether a prediction outside the data range is reliable, state clearly: "No, this is extrapolation outside the range of given data, so the model may not be valid."