Area Between Two Z-Scores Calculator
Enter two Z-scores to find the area (probability) between them under the standard normal distribution curve.
Understanding how to find the area between two z-scores is a fundamental skill in statistics, especially in topics involving the normal distribution. This concept allows you to determine probabilities, percentages, and understand where data points lie in relation to the population mean.
This detailed post will cover:
- What z-scores represent
- The concept of area under the normal curve
- Step-by-step approach to finding the area between two z-scores
- Using z-tables for area calculations
- Practical examples
- Use of calculators and software
- Common scenarios and tips
1. What Are Z-Scores?
A z-score (or standard score) measures how many standard deviations a raw score x is from the mean μ of a population:z=x−μσz=σx−μ
Where:
- xx = raw score
- μμ = mean of the dataset
- σσ = standard deviation
- Positive z-scores indicate values above the mean.
- Negative z-scores indicate values below the mean.
This standardization puts values on a common scale, which is why it’s crucial in probability calculations and hypothesis testing.
2. Understanding Area Under the Normal Distribution Curve
The normal distribution, or bell curve, shows the probability density of a continuous random variable. The total area under this curve is 1 (or 100%).
- The area under the curve between two z-scores corresponds to the probability that a value lies between those two z-scores.
- For example, the area between z = -1 and z = 1 covers about 68.27% of the total distribution, reflecting the empirical rule.
3. Why Calculate the Area Between Two Z-Scores?
Finding this area helps you:
- Determine the likelihood/probability of a data point falling within a range
- Compute percentile scores
- Conduct hypothesis testing in statistics
- Interpret confidence intervals
4. How to Find the Area Between Two Z-Scores (Step-by-Step)
Step 1: Identify Your Z-Scores
Let’s say you want the area between two z-scores, z1z1 and z2z2, where z1<z2z1<z2.
Example: z1=−0.89z1=−0.89, z2=0.18z2=0.18
Step 2: Find Areas Corresponding to Each Z-Score
You can find the area to the left of each z-score from the standard z-table or using a calculator. These values represent cumulative probabilities P(Z<zi)P(Z<zi).
- For z1=−0.89z1=−0.89, cumulative area = 0.1867
- For z2=0.18z2=0.18, cumulative area = 0.5714
Step 3: Calculate the Area Between
Area between z1 and z2=P(Z<z2)−P(Z<z1)Area between z1 and z2=P(Z<z2)−P(Z<z1)
Using the example values:0.5714−0.1867=0.38470.5714−0.1867=0.3847
So approximately 38.47% of the data lies between z=−0.89z=−0.89 and z=0.18z=0.18.
5. Using the Z-Table
The z-table lists the area under the curve to the left of a specific z-score.
Example: Find area between z=−0.46z=−0.46 and z=−0.04z=−0.04
Look up absolute values (because the normal distribution is symmetric):
| z | Area to Left |
|---|---|
| 0.46 | 0.6772 |
| 0.04 | 0.5160 |
Calculate area between:0.6772−0.5160=0.16120.6772−0.5160=0.1612
So 16.12% of data lies between these z-scores.
6. Calculating Area When Both Z-Scores Are on One Side of the Mean
The method is the same: find cumulative probabilities and subtract the smaller from the larger.
Note: Use absolute values to lookup in the z-table when dealing with negative z-scores.
7. Calculating Area When Z-Scores Lie on Opposite Sides of the Mean
Example: between z=−1.44z=−1.44 and z=1.44z=1.44:
Lookup cumulative areas:
- P(Z<1.44)=0.9251P(Z<1.44)=0.9251
- P(Z<−1.44)=0.0749P(Z<−1.44)=0.0749
Area between:0.9251−0.0749=0.85020.9251−0.0749=0.8502
So about 85.02% of data falls between those points.
8. Practical Examples
| Lower Z-Score | Upper Z-Score | Area Between (Probability %) |
|---|---|---|
| -1.00 | 1.00 | 68.27 |
| -2.00 | 2.00 | 95.45 |
| -3.00 | 3.00 | 99.73 |
| -0.5 | 0.5 | 38.29 |
| -1.96 | 1.96 | 95.00 |
9. When Raw Scores Are Given Instead of Z-Scores
- Convert raw scores x1,x2x1,x2 to z-scores using:
z=x−μσz=σx−μ
- Use same process as above with z-values.
10. Tools and Calculators
To avoid tedious lookup and calculation, many websites and software provide automated calculators:
- Input the two z-scores
- Get the area/probability between them instantly
This is helpful for negative scores, decimals, or when high precision is needed.
11. Important Tips and Notes
- The total area under the curve is 1 or 100%.
- Symmetry means P(Z<−a)=1−P(Z<a)P(Z<−a)=1−P(Z<a).
- The z-table provides cumulative area from far left up to z.
- For positive z, table gives area to left; for negative, use symmetry.
- In practice, for quick estimations, memorizing key percentages helps (e.g., 68%, 95%, 99.7%).
12. Summary Steps to Find Area Between Two Z-Scores
| Step | Action |
|---|---|
| 1 | Compute or identify z-scores z1,z2z1,z2 |
| 2 | Use z-table or calculator to find P(Z<z1)P(Z<z1) and P(Z<z2)P(Z<z2) |
| 3 | Subtract smaller from larger: P(Z<z2)−P(Z<z1)P(Z<z2)−P(Z<z1) |
| 4 | Interpret result as probability or percentage area under curve |
13. Visualizing Area Between Two Z-Scores
The area between two z-scores is represented as the shaded region under the bell curve bounded by vertical lines at z1z1 and z2z2.
