The Normal Distribution Table

1. What the Normal Distribution Table Actually Gives You 🧠

Okay — let’s start from the most important fact. The table only gives left-tail probabilities:

P(Z < z)

That’s it.
No right-tail values.
No two-tail values.
No between-region values.

Everything the exam throws at you is built from this one foundational idea 📏. Once you fully accept that, the whole topic becomes a matter of rearranging what you already know.

Examiners deliberately craft questions where the direction switches — “greater than”, “between”, “less than negative value” — to see whether you truly understand this.

2. How to Read the Table Without Confusion ⚙️

The table is arranged in two parts:

• Rows → give the first decimal place (e.g., 1.2)
  
  
• Columns → give the second decimal place (e.g., 0.04)
  
  

So for Z = 1.24:

• Find row 1.2
  
  
• Move across to column 0.04
  
  
• That cell gives (P(Z < 1.24))
  
  

The most common student mistake ❗ is reading the wrong column (e.g., 0.06 instead of 0.04). Edexcel’s examiner reports mention this almost every year.

Expert fix:
Finger on the row. Pen on the column. Meet in the middle 🧠.
It sounds silly, but it prevents 90% of reading errors.

3. Why We Standardise Every Time 📘

Before using the table, we convert x into a Z-value:

z = \frac{x – \mu}{\sigma}

This shifts all normal distributions into the standard normal distribution (mean 0, sd 1). That’s the distribution the table is built for.

AQA often gives a method mark purely for showing the standardisation step. OCR does too. Forgetting to standardise ❗ — or doing it incorrectly — immediately costs marks.

Key takeaway:
If the question doesn’t already give you Z, then you must standardise.

4. Worked Example: (P(Z < 1.24)) ⚙️

This is the friendliest type of question because it matches the table directly.

• Row: 1.2
  
  
• Column: 0.04
  
  
• Table value: 0.8925
  
  

So:

P(Z < 1.24) = 0.8925

A quick sense-check 🧠:
1.24 is slightly more than one standard deviation above the mean, so almost the entire left-hand side of the distribution is shaded. Roughly 89% fits perfectly.

5. Worked Example: (P(Z > 1.90)) ❗

This is where the table’s limitation matters.

The table only gives:

P(Z < 1.90) = 0.9713

But we want the right tail:

P(Z > 1.90) = 1 – 0.9713 = 0.0287

So 2.87% lies above 1.90.

This matchup — left-tail value vs required right-tail value — appears frequently in AQA and OCR.

Mark scheme trigger phrase:
“M1 for using 1 − P(Z < z)”.

6. Worked Example: (P(-0.40 < Z < 1.60)) 🧠

Between-values questions seem more intimidating until you break them into simple parts:

Upper bound:

P(Z < 1.60) = 0.9452

Lower bound:

P(Z < -0.40) = 0.3446

Subtract:

0.9452 – 0.3446 = 0.6006

Meaning:

P(-0.40 < Z < 1.60) = 0.6006

About 60% of values lie in this region. OCR uses this structure often to check fluency 📘.

7. Negative Z-values: The Symmetry Trick ⚙️

Students sometimes panic when they see negative Z-values because the table usually starts at 0.00. But the Normal Distribution is symmetric:

P(Z < -a) = 1 – P(Z < a)

So Z = –0.40 is handled using the positive 0.40 line.

Students who forget symmetry ❗ often waste time searching or guessing.

8. Why Sketching is Still Worth Doing 🧠

Even though this blog doesn’t use diagrams (Statistics → no diagram), you should sketch every probability region during an exam.

Benefits of sketching:

• Stops direction errors (“greater than” vs “less than”)
  
  
• Helps interpret between-values
  
  
• Makes right-tail probability decisions clearer
  
  
• Prevents sign mistakes with negative Z-values
  
  

Edexcel examiners have repeatedly commented that “candidates who sketched were significantly more accurate.”

Sketching is exam superpower, not decoration ⚙️.