Standardisation of a Normal Variable & Using the Standard Normal Table
Hello! Welcome to your study notes for one of the most useful topics in M1 Statistics. Don't worry if words like "normal distribution" or "standardisation" sound a bit scary. By the end of these notes, you'll see they are just simple tools to help us make sense of the world around us.
What will we learn? We'll learn a clever trick called standardisation. Think of it like converting different currencies into one common currency (like Hong Kong Dollars) to easily compare them. We do this with data to find probabilities using a single, powerful tool: the standard normal table.
Why is it important? From your exam scores and heights to the weight of bags of chips, many things in life follow a normal distribution. This topic gives you the key to unlock probabilities for ALL of them! Let's get started.
Recap: What is a Normal Distribution Anyway?
Remember the bell-shaped curve? That's the signature look of a normal distribution. It describes how data is spread out for many real-life situations.
A normal distribution is defined by two key pieces of information:
- Mean (\(\mu\)): The average value, which is right at the center and the peak of the bell curve.
- Standard Deviation (\(\sigma\)): A measure of how spread out the data is. A small \(\sigma\) means the data is tightly packed around the mean (a tall, skinny bell). A large \(\sigma\) means the data is very spread out (a short, wide bell).
We write this as \(X \sim N(\mu, \sigma^2)\), which reads "the variable X is normally distributed with mean \(\mu\) and variance \(\sigma^2\)". (Remember, variance is just standard deviation squared!)
The Big Problem
Imagine we have the exam scores for HKDSE Maths M1, which might have a mean of 65 and a standard deviation of 10. Then we have the scores for HKDSE Physics, with a mean of 60 and a standard deviation of 15. These are two different normal distributions. There are infinitely many possible combinations of \(\mu\) and \(\sigma\)! How could we possibly calculate probabilities for all of them? We can't have a separate probability table for every single one. We need a "one-size-fits-all" solution.
Quick Review: Key Properties of a Normal Distribution
- It's bell-shaped and perfectly symmetrical around the mean (\(\mu\)).
- The mean, median, and mode are all equal and located at the center.
- The total area under the curve is exactly 1 (or 100%). This is super important because area represents probability!
The Star of the Show: The Standard Normal Distribution
To solve our "too many distributions" problem, mathematicians created a special, reference distribution called the standard normal distribution. It's the superstar that all other normal distributions can be compared to.
The standard normal distribution has fixed properties:
- Mean (\(\mu\)) is always 0.
- Standard Deviation (\(\sigma\)) is always 1.
We use the letter Z to represent a standard normal variable. So, we write: \(\bf{Z \sim N(0, 1)}\)
Because this distribution is "standard," we have a special table of values for it—the standard normal table. This table allows us to find the area (probability) for any value of Z.
Key Takeaway
Instead of working with countless different normal distributions (\(N(\mu, \sigma^2)\)), our goal is to convert any of them into the ONE standard normal distribution (\(N(0, 1)\)). This conversion process is called standardisation.
The Magic Formula: How to Standardise (The Z-score)
To convert any value (X) from a normal distribution into its equivalent value (Z) on the standard normal distribution, we use the Z-score formula. This is your most important formula for this chapter!
\(\bf{Z = \frac{X - \mu}{\sigma}}\)Breaking Down the Formula:
- Z: This is the Z-score you are calculating. It tells you how many standard deviations (\(\sigma\)) your original value (X) is away from the mean (\(\mu\)).
- X: Your original data point or value of interest (e.g., an exam score of 75).
- \(\mu\): The mean of your original distribution (e.g., the class average was 65).
- \(\sigma\): The standard deviation of your original distribution (e.g., the score spread was 5).
An Everyday Analogy: Comparing Test Scores
Imagine you and your friend take different M1 mock exams.
- Your Test: You scored 80. The class mean (\(\mu\)) was 70 and the standard deviation (\(\sigma\)) was 10.
- Friend's Test: Your friend scored 85. Their class mean (\(\mu\)) was 75 and the standard deviation (\(\sigma\)) was 5.
Who performed better relative to their own class? Let's standardise the scores by finding the Z-score for each of you.
Your Z-score: \(Z = \frac{80 - 70}{10} = \frac{10}{10} = \bf{+1.0}\) This means you scored exactly 1 standard deviation above your class average.
Friend's Z-score: \(Z = \frac{85 - 75}{5} = \frac{10}{5} = \bf{+2.0}\) This means your friend scored a whopping 2 standard deviations above their class average.
Conclusion: Even though your friend's score of 85 is only 5 marks higher than your 80, their Z-score is much higher. They performed significantly better compared to their classmates. The Z-score gives us a fair way to compare!
What does a Z-score mean?
- A positive Z-score means the value (X) is above the average (\(\mu\)).
- A negative Z-score means the value (X) is below the average (\(\mu\)).
- A Z-score of 0 means the value (X) is exactly the average (\(\mu\)).
Using the Standard Normal Table
Once you have a Z-score, you can use the standard normal table (which is printed in your exam paper) to find probabilities.
IMPORTANT — read this twice! The table used in the HKDSE M1 exam gives the area between 0 and z. We call this \( \bf{A(z) = P(0 < Z < z)} \). It does NOT give the area to the left of z. The table starts at \( A(0) = 0 \) and grows towards 0.5 as z gets large, and it only lists positive z values — for negative values you use symmetry.
Because the curve is symmetrical and the total area is 1, each half of the curve has area 0.5. That "0.5 baseline" is the key to every calculation below.
How to Read the Table (to find \(A(1.34)\))
- Split the Z-score: Split 1.34 into "1.3" and "0.04".
- Find the Row: Look down the left-most column to find the row for 1.3.
- Find the Column: Look along the top row to find the column for .04.
- Find the Intersection: The value where the row and column meet is the area from 0 to 1.34. You should find \( A(1.34) = 0.4099 \).
So \( P(0 < Z < 1.34) = 0.4099 \). If you want the area to the left of 1.34, add the left half of the curve: \( P(Z < 1.34) = 0.5 + A(1.34) = 0.5 + 0.4099 = 0.9099 \).
Handling Different Types of Probabilities
Every question becomes a little picture of areas. Sketch the curve, shade the region you want, then build it from 0.5 and A(z) values.
1. Less Than probability: \( P(Z < z) \) for positive z
Formula: \( \bf{P(Z < z) = 0.5 + A(z)} \)
Example: \( P(Z < 1.34) = 0.5 + 0.4099 = \bf{0.9099} \)
2. Greater Than probability: \( P(Z > z) \) for positive z
The right half has area 0.5, and \(A(z)\) is the part of it between 0 and z. What is left over is the tail.
Formula: \( \bf{P(Z > z) = 0.5 - A(z)} \)
Example: \( P(Z > 1.34) = 0.5 - 0.4099 = \bf{0.0901} \)
3. Negative values: \( P(Z < -z) \)
The curve is symmetrical! The area to the left of \(-z\) equals the area to the right of \(+z\).
Formula: \( \bf{P(Z < -z) = P(Z > z) = 0.5 - A(z)} \)
Example: \( P(Z < -1.34) = 0.5 - A(1.34) = 0.5 - 0.4099 = \bf{0.0901} \)
4. "Between" two values: \( P(a < Z < b) \)
- Both on the same side (0 < a < b): \( \bf{P(a < Z < b) = A(b) - A(a)} \)
- Straddling zero (a < 0 < b): the region is made of two pieces, one on each side of 0: \( \bf{P(a < Z < b) = A(|a|) + A(b)} \)
Example: Find \(P(-1.0 < Z < 1.5)\)
The region straddles zero, so add the two pieces:
\( A(1.0) = 0.3413 \) and \( A(1.5) = 0.4332 \)
\( P(-1.0 < Z < 1.5) = 0.3413 + 0.4332 = \bf{0.7745} \)
Common Mistakes to Avoid
- Reading the table value as \(P(Z < z)\): the number printed in the DSE table is only the area from 0 to z. For "less than", you must add 0.5. This is the single most common M1 table error!
- Forgetting the 0.5 baseline: for tails, subtract \(A(z)\) FROM 0.5 — never from 1 directly.
- Symmetry errors: sketching a quick bell curve makes negative Z-scores much easier to handle.
Putting It All Together: A Full Example
The weights of apples from an orchard are normally distributed with a mean of 150g (\(\mu\)) and a standard deviation of 12g (\(\sigma\)). Find the probability that a randomly chosen apple weighs between 140g and 165g.
Step 1: Write down the information and the goal.
We have a normal distribution: \(X \sim N(150, 12^2)\).
We want to find \( \bf{P(140 < X < 165)} \).
Step 2: Standardise BOTH X-values into Z-scores.
For X = 140: \( Z_1 = \frac{140 - 150}{12} = \frac{-10}{12} \approx -0.83 \)
For X = 165: \( Z_2 = \frac{165 - 150}{12} = \frac{15}{12} = 1.25 \)
Step 3: Rewrite the problem in terms of Z.
\( P(140 < X < 165) \) is the same as \( \bf{P(-0.83 < Z < 1.25)} \)
Step 4: The region straddles zero — add the two pieces.
From the table: \( A(0.83) = 0.2967 \) and \( A(1.25) = 0.3944 \)
\( P(-0.83 < Z < 1.25) = A(0.83) + A(1.25) \)
Step 5: Calculate the final answer.
\( 0.2967 + 0.3944 = \bf{0.6911} \)
Conclusion: The probability that a randomly chosen apple weighs between 140g and 165g is approximately 0.6911 (or 69.11%).
Summary and Key Takeaways
You've made it! Let's quickly review the main ideas.
- The Why: We standardise to turn any normal distribution \(N(\mu, \sigma^2)\) into the one Standard Normal Distribution \(N(0, 1)\), so we can use a single table to find probabilities.
- The How (The Z-score Formula): \( \bf{Z = \frac{X - \mu}{\sigma}} \) This formula is your key to everything.
- The Tool (The HKDSE Table): The exam table gives \( \bf{A(z) = P(0 < Z < z)} \) — the area between 0 and z. Build everything from it and the 0.5 half-curve baseline:
- \( \bf{P(Z < z) = 0.5 + A(z)} \)
- \( \bf{P(Z > z) = 0.5 - A(z)} \)
- \( \bf{P(Z < -z) = 0.5 - A(z)} \) (symmetry)
- \( \bf{P(a < Z < b) = A(b) - A(a)} \) (same side) or \( \bf{A(|a|) + A(b)} \) (straddling 0)
Always show your steps clearly: write the formula, substitute the values, find the Z-score, sketch the curve, and then build the probability from the table. You can do it!