Making estimates Revision Notes for AQA GCSE Statistics

Making estimates

What are estimates in statistics?

When working with data, we often can't measure an entire population due to time, cost, or practical constraints. Instead, we collect information from a representative sample and use key statistics like the mean, median, range, and interquartile range to make educated predictions about the whole population. This process is called making estimates, and it's a fundamental skill in data analysis.

Understanding quartiles and their meanings

To make accurate estimates, you need to understand how quartiles work. Quartiles are values that divide your dataset into four equal parts, each containing 25% of the data.

infoNote

How quartiles divide your data:

50% of values lie below the median, while 50% lie above it
25% of values fall below the lower quartile (Q1)
25% of values fall above the upper quartile (Q3)
50% of values sit between the lower and upper quartiles

This creates a five-number summary that gives us a complete picture of how data is spread out.

Step-by-step method for making population estimates

When you want to estimate how many individuals in a population possess a particular characteristic, follow this systematic approach:

Step 1: Identify the relevant percentage from your sample

Examine your sample data to determine what proportion demonstrates the characteristic you're investigating. This might be expressed as a quartile position or percentile.

Step 2: Convert percentage to decimal form

Transform your percentage into decimal format (25% becomes 0.25, 50% becomes 0.5, etc.).

Step 3: Apply to total population

Use the fundamental estimation formula:

$\text{Estimated number} = \text{Sample percentage (as decimal)} \times \text{Total population size}$

Step 4: Check your answer makes sense

Verify that your result is reasonable given the context and population size.

Detailed worked example: Hospital blood pressure data

lightbulbExample

Worked Example: Hospital Blood Pressure Estimation

Given information:

Sample shows five-number summary of blood pressure readings
Lower quartile = 112.4 mmHg
Median = 116.7 mmHg
Total daily UK hospital admissions = 41,500 patients

Part (a): What proportion had systolic blood pressure less than 112.4?

Since 112.4 represents the lower quartile, we know that exactly 25% of the sample recorded readings below this value.

Part (b): Estimate UK patients with readings less than 112.4

Using our estimation method:

Sample percentage = 25% = 0.25
Calculation: $0.25 \times 41,500 = 10,375$ patients

Part (c): Estimate UK patients with readings greater than 116.7

Since 116.7 is the median:

50% of patients had readings above this value
Calculation: $0.5 \times 41,500 = 20,750$ patients

Working with percentile tables

Percentile tables provide detailed breakdowns of data distribution at various percentage points. Understanding these tables is crucial for making accurate estimates.

This weight percentile table for 18-month-old boys shows that if you select the 25th percentile (10.5 kg), exactly 25% of boys in the sample weigh less than 10.5 kg.

Similarly, this income distribution table reveals that 25% of UK taxpayers earned less than £15,500, while 75% earned less than £34,500 in the specified tax year.

Advanced calculation techniques

Finding populations between two values

When estimating numbers falling between two percentile points:

Identify the percentile for each boundary value
Calculate the difference between these percentiles
Apply this percentage difference to your total population

infoNote

Example calculation between percentiles:

If you need the number between the 25th and 75th percentiles:

$75\% - 25\% = 50\%$ of the population
Multiply $0.50$ by total population size

Working with percentile ranges

Remember that percentiles are cumulative - the 60th percentile means 60% of values fall below that point, not that 60% fall exactly at that value.

Common exam techniques and pitfalls

infoNote

Essential exam strategies:

Always read whether the question asks for a number of people or a percentage
Double-check your percentage-to-decimal conversions
Ensure your final answer is appropriate for the context (you can't have 0.7 of a person!)
Show your working clearly, including the formula you're using

chatImportant

Typical mistakes to avoid:

Confusing "less than" with "greater than" when interpreting quartiles
Using percentages instead of decimals in calculations
Misinterpreting what percentiles represent
Forgetting to round answers sensibly

Quick verification checks

Your results should pass these logical tests:

Your percentages should logically add up (below median + above median = 100%)
Estimated numbers should be reasonable compared to total population
Lower percentiles should give smaller estimated numbers than higher percentiles

bookmarkSummary

Key Points to Remember:

Use representative samples to predict population characteristics by applying sample percentages to total population sizes
Quartiles split data into four equal 25% sections, with the median creating a perfect 50-50 division
The core estimation formula is: $\text{Sample percentage (as decimal)} \times \text{Total population} = \text{Estimated number}$
Percentiles are cumulative - the 75th percentile means 75% of all values fall below that measurement
Always round final answers appropriately based on real-world context and check they make logical sense

Making estimates (AQA GCSE Statistics): Revision Notes