Mathematics | Representation and summary of data

Unit S1: Representation and Summary of Data

Welcome to the fascinating world of Statistics! This chapter is your foundation stone. We’re going to learn how to take messy raw numbers and turn them into clear, insightful stories using graphs and key summary calculations.

Why is this important? Because simply looking at a long list of numbers tells us very little. By representing and summarizing data effectively, we can spot trends, compare groups, and make informed decisions—skills essential not just for your exams, but for life! Don't worry if concepts like Variance seem abstract; we will break them down step-by-step. Let’s get started!

Section 1: Types of Data (The Building Blocks)

Before we can analyze data, we must know what kind of data we have. Data generally falls into two major categories:

1. Qualitative vs. Quantitative Data

• Qualitative Data: Describes characteristics or categories. It is non-numerical.
  Example: Eye color, car model, favorite flavor.
• Quantitative Data: Data that consists of numbers and can be measured or counted. This is the focus of most of S1.

2. Discrete vs. Continuous Data (Focusing on Quantitative)

• Discrete Data: Data that can only take specific, fixed values (usually whole numbers). It often arises from counting.
  Analogy: Discrete data is like people in a room—you can’t have 3.5 people.
  Example: The number of cars passing a point, shoe size (UK sizes are specific steps).
• Continuous Data: Data that can take any value within a given range. It arises from measuring.
  Analogy: Continuous data is like sand—you can always find a value between two others.
  Example: Height, weight, temperature, time taken to run a race.

Quick Tip: Data collected for continuous variables (like height) is often recorded using class intervals (e.g., 170cm to 180cm). Always check the boundaries!

Section 2: Visual Representation of Data

Graphs help us see the overall shape (the distribution) of the data.

1. Stem and Leaf Diagrams

These diagrams preserve the raw data while presenting it in an organized format. They are great for small to medium-sized datasets.

• The Stem shows the larger place value (e.g., tens, hundreds).
• The Leaf shows the smallest place value (e.g., units, tenths).
• The leaves must always be in numerical order, starting closest to the stem.
• Crucial Step: You must include a Key! Without a key, the diagram is meaningless.
  Example Key: 2 | 5 means 25.

Did You Know? We use back-to-back stem and leaf plots to easily compare two related datasets (e.g., test scores of boys vs. girls).

2. Histograms (For Continuous Data)

Histograms are used for continuous, grouped data. This is a common exam topic, so pay close attention!

The Golden Rule of Histograms: The area of the bar must be proportional to the frequency (the number of observations) in that class.

Since the class widths are often unequal, we cannot just plot frequency against the class interval (as we would for a bar chart). We must calculate Frequency Density for the y-axis.

Frequency Density \( = \frac{\text{Frequency}}{\text{Class Width}} \)

Step-by-Step Construction:

1. Determine the Class Width for each group (\( \text{Upper Boundary} - \text{Lower Boundary} \)).
2. Calculate the Frequency Density for each group.
3. Plot the class interval on the horizontal (x) axis.
4. Plot the Frequency Density on the vertical (y) axis.
5. Draw rectangles where the area is proportional to the frequency.

Common Mistake to Avoid: When dealing with grouped continuous data (e.g., 10-19, 20-29), make sure you use the true class boundaries (e.g., 9.5 to 19.5, 19.5 to 29.5) to calculate the correct Class Width (which would be 10 in this case).

3. Cumulative Frequency Diagrams (Ogive)

A cumulative frequency graph shows the running total of frequencies. It is crucial for estimating the median and quartiles from grouped data.

Step-by-Step Construction:

1. Calculate the Cumulative Frequency (CF) by adding the frequencies sequentially.
2. Plot the CF values against the upper boundary of each class interval.
3. The graph should start at (Lower boundary of the first class, 0).
4. Connect the points with a smooth curve (not straight lines).

Quick Tip: The highest point on the y-axis (the final cumulative frequency) should equal the total number of observations, \(n\).

Section 3: Measures of Central Tendency (Location)

These statistics tell us about the 'centre' or typical value of the dataset.

1. The Mean (\( \bar{x} \))

The mean is the arithmetic average. It uses every data point and is sensitive to extreme values (outliers).

• Raw Data Mean: $$ \bar{x} = \frac{\sum x}{n} $$ Where \( \sum x \) is the sum of all data points and \(n\) is the number of data points.
• Frequency Table Mean: $$ \bar{x} = \frac{\sum fx}{\sum f} $$ Where \(f\) is the frequency and \(x\) is the data value.
• Grouped Data Mean (Estimation): We must assume that all values in a class are concentrated at the midpoint (\(m\)) of that class. $$ \bar{x} \approx \frac{\sum fm}{\sum f} $$

2. The Median

The median is the middle value when the data is arranged in order. It is unaffected by outliers.

• Raw Data Median:
  First, order the data. The position of the median is usually given by \( \frac{n+1}{2} \).
• Grouped Continuous Data Median (Interpolation):
  We use the cumulative frequency distribution to estimate the median, usually found at the position \( \frac{n}{2} \).
  Process: Locate the median position (\( \frac{n}{2} \)) on the vertical (CF) axis. Draw a horizontal line to the curve, then drop vertically to the horizontal (data value) axis to read the estimated median.

Analogy: The Median is the "safe" measure. If someone throws an extremely high number into your dataset (an outlier), the Mean gets pulled dramatically towards it, but the Median stays relatively stable.

3. The Mode (or Modal Class)

The mode is the value that occurs most frequently.

• For raw or discrete data, it is the actual value that appears most often.
• For grouped data, we identify the Modal Class (the class with the highest frequency density).

Section 4: Measures of Dispersion (Spread)

These statistics tell us how spread out or varied the data is.

1. Range and Interquartile Range (IQR)

• Range: \( \text{Maximum Value} - \text{Minimum Value} \). Simple, but heavily affected by outliers.
• Quartiles: Divide the data into four equal parts.
  • \(Q_1\) (Lower Quartile): 25% of data is below this point.
  • \(Q_2\) (Median): 50% of data is below this point.
  • \(Q_3\) (Upper Quartile): 75% of data is below this point.
• Interquartile Range (IQR): \( \text{IQR} = Q_3 - Q_1 \). This measures the spread of the middle 50% of the data and is resistant to outliers.

Finding Quartiles for Grouped Data: Similar to the median, use the cumulative frequency curve.

• \(Q_1\) is found at position \( \frac{n}{4} \).
• \(Q_3\) is found at position \( \frac{3n}{4} \).

2. Variance and Standard Deviation

The Variance (\( \sigma^2 \)) and Standard Deviation (\( \sigma \)) are the most robust measures of spread, as they use all the data points to measure the deviation from the mean.

Standard Deviation (\( \sigma \)) is simply the square root of the Variance. It is preferred because it is measured in the same units as the original data.

Formulas for Calculation: (You should be familiar with both the definition formula and the computational formula.)

A. Raw Data Formulas (n observations):

Definition Formula (variance): $$ \sigma^2 = \frac{\sum (x - \bar{x})^2}{n} $$ (This means: find the difference from the mean, square it, sum them up, and divide by \(n\).)

Computational Formula (variance): (Easier for calculations, especially without a calculator’s Stats Mode.) $$ \sigma^2 = \frac{\sum x^2}{n} - (\bar{x})^2 $$

B. Frequency Table Formulas (Grouped or Ungrouped):

Computational Formula (variance): $$ \sigma^2 = \frac{\sum fx^2}{\sum f} - (\bar{x})^2 $$ (Where \( \sum f \) is the total frequency, \(n\).)

Memory Aid: For the standard deviation computational formula, remember: "The Mean of the Squares minus the Square of the Mean."

\( \sigma = \sqrt{\frac{\sum x^2}{n} - (\bar{x})^2} \)

Section 5: Summary Diagrams and Outliers

1. Box Plots (Box and Whisker Diagrams)

A box plot provides a quick visual summary of the five key statistics (the Five-Number Summary):

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

Box plots are extremely useful for comparing the spread and location of two or more datasets visually.

2. Identifying Outliers

An Outlier is an extreme value that lies far away from the other data points. We need a rigorous method to decide if a value is genuinely an outlier.

In S1, we use the Interquartile Range (IQR) method. A data point \(x\) is considered an outlier if it falls outside the following fences:

• Lower Fence: \( Q_1 - 1.5 \times \text{IQR} \)
• Upper Fence: \( Q_3 + 1.5 \times \text{IQR} \)

Example: If a value is smaller than the Lower Fence OR larger than the Upper Fence, it is marked as an outlier (often plotted as a cross or asterisk on a box plot).

Remember: When drawing a box plot that includes outliers, the "whiskers" extend only to the highest and lowest values that are not outliers.

3. Effect of Coding Data

Sometimes, to simplify calculations, we "code" the data using a linear transformation: \( y = \frac{x - a}{b} \), where \(a\) and \(b\) are constants.

• Measures of Location (\(\bar{x}, Q_2, Q_1, Q_3\)): These are affected by both addition/subtraction (\(a\)) and multiplication/division (\(b\)). If \( x \to y \), then \( \bar{x}_y = \frac{\bar{x}_x - a}{b} \).
• Measures of Spread (Range, IQR, \(\sigma\)): These are ONLY affected by multiplication or division (\(b\)). Adding/subtracting \(a\) shifts the data but doesn't change the spread.
  If \( y = \frac{x - a}{b} \), then \( \sigma_y = \frac{\sigma_x}{|b|} \) or \( \text{IQR}_y = \frac{\text{IQR}_x}{|b|} \).

Analogy for Coding: If everyone's score in a test goes up by 10 points (\(x+10\)), the average goes up by 10, but the spread (standard deviation) remains the same because everyone is shifted equally.