Introduction: Bringing Order to Chaos
Imagine walking into a school canteen and asking \(50\) students what their favourite snack is. If you just write down their answers on a scrap of paper as they shout them out, you will end up with a messy list of words. This is called raw data.
In Statistics, raw data can be overwhelming and difficult to interpret. That is where tables, tally charts, and two-way tables come in! These tools allow us to organise, summarise, and display data clearly so that we can spot patterns, calculate statistics, and answer real-world questions quickly.
Don't worry if this seems a bit detailed at first — we will break down each method step by step.
1. Tally Charts and Ungrouped Frequency Tables
What is a Tally Chart?
A tally chart is a simple way of recording data as you collect it. Instead of writing numbers, you make small marks called tallies.
To make tallies easy to count at the end:
• Draw single vertical marks for the first four items: \(|\), \(||\), \(|||\), \(||||\)
• For the \(5^{\text{th}}\) item, draw a diagonal line across the first four: \( \rlap{////}{||||} \) (this makes a neat bundle of \(5\))
• Counting in fives at the end is much faster and helps prevent counting errors!
What is Frequency?
Frequency simply means "how many times something happens". The total frequency is represented by the letter \(n\) or the symbol \(\sum f\) (where \(\sum\) means "sum of" and \(f\) means frequency).
Example: Ungrouped Frequency Table
A survey asks \(20\) students how many pets they own. The raw responses are:
\(1, 0, 2, 1, 3, 1, 0, 0, 2, 1, 1, 4, 2, 1, 0, 1, 3, 2, 1, 0\)
We can organise this into an ungrouped frequency table (used for discrete data with a small range of values):
Number of Pets | Tally | Frequency (\(f\))
\(0\) pets | \( \rlap{////}{||||} \) | \(5\)
\(1\) pet | \( \rlap{////}{||||} \text{ } || \) | \(7\)
\(2\) pets | \(||||\) | \(4\)
\(3\) pets | \(||\) | \(2\)
\(4\) pets | \(|\) | \(1\)
Total (\(\sum f\)) | | \(20\)
Top Tip: Always sum your frequency column at the end. If the total does not match the number of people surveyed (here, \(20\)), you know you have missed a piece of data!
Key Takeaways for Tally Charts:
• Group tallies in bundles of \(5\).
• Frequency (\(f\)) is the total count for each category.
• The sum of frequencies (\(\sum f\)) must equal the total sample size.
2. Grouped Frequency Tables
Why Group Data?
When dealing with a large range of numbers or continuous data (such as heights, weights, or times), an ungrouped table would have far too many individual rows. Instead, we group the data into class intervals.
Understanding Class Intervals and Inequalities
Class intervals must be chosen carefully so that every single data value fits into exactly one group, with no gaps and no overlaps.
For continuous data, we often use inequality notation:
• \(0 \le t < 10\) means "\(t\) is greater than or equal to \(0\), but strictly less than \(10\)".
• A value of \(9.9\) goes into \(0 \le t < 10\).
• A value of \(10.0\) goes into the next class: \(10 \le t < 20\).
Example: Time Spent on Homework
The time, \(t\) minutes, spent by \(30\) students on an assignment is recorded:
Time (\(t\) minutes) | Frequency (\(f\))
\(0 \le t < 15\) | \(4\)
\(15 \le t < 30\) | \(11\)
\(30 \le t < 45\) | \(10\)
\(45 \le t < 60\) | \(5\)
Total (\(\sum f\)) | \(30\)
Important Properties of Grouped Data:
• Class width: The size of the interval. For \(15 \le t < 30\), the class width is \(30 - 15 = 15\).
• Mid-interval value (midpoint): The middle of the class, found by adding the lower and upper boundaries and dividing by \(2\).
For \(15 \le t < 30\), the midpoint is \(\frac{15 + 30}{2} = 22.5\).
Common Mistake to Avoid: When data is grouped, you lose the original individual values. For example, from the table above, you know \(11\) students took between \(15\) and \(30\) minutes, but you do not know their exact times!
Key Takeaways for Grouped Tables:
• Use grouped tables for continuous data or large ranges of discrete data.
• Watch the inequality signs carefully so values are not counted twice.
• Class intervals must not overlap.
3. Two-Way Tables
What is a Two-Way Table?
A two-way table displays data that involves two categorical variables simultaneously. One variable is shown across the rows (horizontally), and the other is shown down the columns (vertically).
For example, if you want to record the Year Group (Year \(11\) or Year \(12\)) and the Transport Method (Bus or Walk), a two-way table is the ideal way to present it.
The Structure of a Two-Way Table
A complete two-way table always includes:
1. Row categories and Row totals
2. Column categories and Column totals
3. A Grand Total in the bottom-right corner (which is the total number of items or people in the entire dataset).
4. Completing Two-Way Tables: The Detective Method
GCSE exam questions often provide a two-way table with several numbers missing. Finding the missing values is like solving a mini puzzle.
The Golden Rules:
• Each row must add up to its Row Total.
• Each column must add up to its Column Total.
• The sum of all row totals MUST equal the sum of all column totals, which gives the Grand Total.
Step-by-Step Strategy:
1. Look for a row or column that has only one missing value.
2. Subtract the known value(s) from the total of that row or column to find the missing number.
3. Write that number in the table.
4. Repeat the process for the next row or column until the table is complete!
Worked Example:
A group of \(60\) students in Years \(11\) and \(12\) were asked whether they study Art or Music.
We are given the following information:
• Total students: \(60\)
• Year \(11\) students: \(32\)
• Year \(12\) students who study Music: \(15\)
• Total students studying Art: \(25\)
Let's fill in the two-way table step by step:
Initial Table:
Group | Art | Music | Total
Year \(11\) | \(?\) | \(?\) | \(32\)
Year \(12\) | \(?\) | \(15\) | \(?\)
Total | \(25\) | \(?\) | \(60\)
Step 1: Find the Total for Year \(12\).
\(\text{Year } 12 \text{ Total} = 60 - 32 = 28\)
Step 2: Find Year \(12\) Art students.
\(\text{Year } 12 \text{ Art} = \text{Year } 12 \text{ Total} - \text{Year } 12 \text{ Music} = 28 - 15 = 13\)
Step 3: Find Year \(11\) Art students.
\(\text{Year } 11 \text{ Art} = \text{Total Art} - \text{Year } 12 \text{ Art} = 25 - 13 = 12\)
Step 4: Find Year \(11\) Music students.
\(\text{Year } 11 \text{ Music} = \text{Year } 11 \text{ Total} - \text{Year } 11 \text{ Art} = 32 - 12 = 20\)
Step 5: Find the Total Music students (and check consistency).
\(\text{Total Music} = 20 + 15 = 35\)
Check grand total: \(\text{Total Art} + \text{Total Music} = 25 + 35 = 60\). It matches!
Completed Table:
Group | Art | Music | Total
Year \(11\) | \(12\) | \(20\) | \(32\)
Year \(12\) | \(13\) | \(15\) | \(28\)
Total | \(25\) | \(35\) | \(60\)
5. Calculating Probabilities from Two-Way Tables
Two-way tables are frequently used in CCEA GCSE Statistics to calculate probabilities. The key is to identify the correct numerator (top number) and denominator (bottom number).
The standard probability formula is:
\(P(\text{Event}) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}\)
Types of Probability Questions:
1. Whole Sample Probability
When selecting a student at random from the entire group, the denominator is the Grand Total.
Example: A student is chosen at random from all \(60\) students. What is the probability that they study Art?
\(P(\text{Art}) = \frac{\text{Total Art}}{\text{Grand Total}} = \frac{25}{60} = \frac{5}{12}\)
2. Specific Intersection Probability
When selecting a student who belongs to both specific categories from the entire group.
Example: What is the probability that a randomly chosen student is in Year \(11\) and studies Music?
\(P(\text{Year } 11 \text{ and Music}) = \frac{20}{60} = \frac{1}{3}\)
3. Conditional / Subgroup Probability ("Given that...")
Be careful! When the question restricts the choice to a specific row or column, the denominator changes to that row total or column total, NOT the grand total.
Example: Given that a chosen student is in Year \(12\), find the probability that they study Music.
• Here, we are only looking at Year \(12\) students (Total = \(28\)).
• Number of Year \(12\) students studying Music = \(15\).
\(P(\text{Music given Year } 12) = \frac{15}{28}\)
Exam Tip: Look out for phrases like "Given that...", "Out of the boys...", or "If a Year 11 student is chosen...". These words tell you that the denominator is a row or column total, not the grand total!
Summary & Revision Checklist
Before moving on to the next chapter, check that you can:
• Accurately record data using tallies in groups of \(5\).
• Construct and interpret both ungrouped and grouped frequency tables.
• Find the midpoint and class width of a class interval.
• Fill in missing values in a two-way table using row and column totals.
• Calculate simple and conditional probabilities directly from a two-way table.