How to Use the Standard Normal Distribution Table (Z-Table) in Solving Problems
The concept of Standard Normal Distribution plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Students often encounter it in statistics, probability, and various entrance exams like JEE and Olympiads. Mastering this distribution enhances problem-solving abilities related to data analysis and interpretation.
What Is Standard Normal Distribution?
A Standard Normal Distribution is defined as a special form of the normal distribution where the mean (average) is 0 and the standard deviation is 1. You’ll find this concept applied in areas such as probability, statistics, and data science. It is represented by a symmetrical, bell-shaped curve called the standard normal curve or bell curve.
Key Formula for Standard Normal Distribution
Here’s the standard formula: \( Z = \frac{X - \mu}{\sigma} \ )
Where:
- \( X \) = value of the variable
- \( \mu \) = mean of the data set
- \( \sigma \) = standard deviation
Why Is Standard Normal Distribution Important?
Standard Normal Distribution helps in finding probabilities and comparing scores from different normal distributions. It makes problem-solving faster by using Z-tables and allows comparison of results in different contexts, such as exam scores or heights. This makes it a vital tool for board exams and Olympiad preparation.
Properties and Graph of Standard Normal Distribution
- Mean (\( \mu \)) = 0, Standard deviation (\( \sigma \)) = 1
- Shape: Symmetrical and bell-shaped
- Total area under the curve = 1
- About 68% of data lies between Z = –1 and Z = 1
- About 95% between Z = –2 and Z = 2
- About 99.7% between Z = –3 and Z = 3
The curve is centered at zero and extends infinitely in both directions, getting closer to the x-axis but never touching it.
Standard Normal Distribution Table (Z-Table)
A standard normal distribution table, or Z-table, gives the area (or probability) to the left of a Z value. It helps find the probability that a statistic is less than a certain value. For quick reference, the leftmost column gives the first two digits of Z, and the top row gives the second decimal place. Use them together to find the area.
| Z | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 |
|---|---|---|---|---|---|---|
| -1.4 | 0.0808 | 0.0793 | 0.0778 | 0.0764 | 0.0749 | 0.0735 |
| 0.0 | 0.5000 | 0.5040 | 0.5080 | 0.5120 | 0.5160 | 0.5199 |
| 1.0 | 0.8413 | 0.8438 | 0.8461 | 0.8485 | 0.8508 | 0.8531 |
| 1.3 | 0.9032 | 0.9049 | 0.9066 | 0.9082 | 0.9099 | 0.9115 |
For the full table and printable version, check the download section or resources on Vedantu.
How to Use the Z-Table
- Calculate the Z-score using the formula: \( Z = \frac{X - \mu}{\sigma} \)
- Find the value of Z in the leftmost column and the second decimal in the top row.
- The intersecting cell gives the probability (area under the curve left of Z).
- If you want area to the right, subtract this number from 1.
Solved Examples on Standard Normal Distribution
Example 1: Suppose the heights of students in a class are normally distributed with mean 165 cm and standard deviation 10 cm. What is the probability that a randomly selected student is taller than 175 cm?
1. Find Z:\( Z = \frac{175 - 165}{10} = 1 \)2. Probability to the left of Z=1 is 0.8413 (from the Z-table).
3. Probability that a student is taller: 1 – 0.8413 = 0.1587
4. Final Answer: There is a 15.87% chance a student is taller than 175 cm.
Example 2: A laptop's battery life is normally distributed, mean = 50 hours, standard deviation = 15 hours. What is the probability the battery lasts between 50 and 70 hours?
1. Z for 50: \( Z = \frac{50 - 50}{15} = 0 \)2. Z for 70: \( Z = \frac{70 - 50}{15} \approx 1.33 \)
3. Area between Z=0 and Z=1.33: From table, left of 1.33 is 0.9082, left of 0 is 0.5. So, 0.9082 – 0.5 = 0.4082
4. Final Answer: 40.82% probability that the battery lasts between 50 and 70 hours.
Try These Yourself
- Find the probability that a value is below Z = –1.2.
- Calculate the Z-score for a test score of 80 (mean 70, SD 5).
- If 68% of data lies between what Z scores?
- Is the curve for standard normal distribution symmetrical?
Frequent Errors and Misunderstandings
- Mixing up “area to the left” with “area to the right” in the Z-table.
- Forgetting to subtract from 1 when finding probability to the right.
- Confusing standard normal distribution (mean 0, SD 1) with any normal distribution.
- Entering wrong values for mean or standard deviation.
- Skipping stepwise calculation of Z-score.
Classroom Tip
Remember: The standard normal distribution is always centered at 0. Use symmetry whenever possible — the area to the right of Z is the same as the area to the left of –Z. Vedantu’s live teachers often use drawing and table lookups to help students practice this quickly.
Relation to Other Concepts
The idea of standard normal distribution connects closely with topics such as Normal Distribution, Z-Score, Standard Deviation, and Mean. Mastering this helps with understanding more advanced statistical concepts and techniques.
Wrapping It All Up
We explored Standard Normal Distribution—from definition and formula to solved examples and common mistakes. Continue practicing with Vedantu to become confident in solving probability problems with this important topic. For further reading and resources, check out Probability Density Function.
FAQs on Standard Normal Distribution Explained with Formula, Table & Examples
1. What is a standard normal distribution in maths?
The standard normal distribution is a specific type of normal distribution. It's characterized by a mean of 0 and a standard deviation of 1. This makes it a crucial tool in statistics and probability, simplifying calculations and comparisons across different datasets.
2. How does the standard normal distribution differ from a regular normal distribution?
A regular normal distribution can have any mean (μ) and standard deviation (σ). The standard normal distribution is a special case where μ = 0 and σ = 1. This standardization allows for easy use of z-tables and simplifies probability calculations.
3. What is the formula for standard normal distribution?
The standard normal distribution doesn't have a unique formula in the same way a regular normal distribution does (using μ and σ). Instead, it uses the z-score formula to standardize values: z = (x - μ) / σ, where 'x' is the raw score, 'μ' is the population mean, and 'σ' is the population standard deviation. When applied to the standard normal distribution, μ becomes 0 and σ becomes 1, simplifying the calculation to z = x.
4. How do you use a z-table to find probabilities?
A z-table (also called a standard normal table) provides the cumulative probability for a given z-score. To use it:
• Locate the z-score in the table (usually given to two decimal places).
• The corresponding value represents the area under the standard normal curve to the left of that z-score. This value is the cumulative probability. This area corresponds to the probability that a randomly selected value will be less than or equal to the z-score.
5. Why is the standard normal distribution important in exams?
The standard normal distribution is frequently tested in statistics and probability exams. Understanding it is crucial for solving problems involving probability calculations, hypothesis testing, and confidence intervals. It simplifies calculations and allows for using the z-table for quick probability lookups.
6. What are the key properties of the standard normal distribution?
Key properties include:
• Mean (μ) = 0
• Standard Deviation (σ) = 1
• Symmetrical about the mean
• Total area under the curve = 1
• The curve approaches, but never touches, the x-axis as it extends to positive and negative infinity.
7. How do you calculate a z-score?
The z-score is calculated using the formula: z = (x - μ) / σ, where:
• x is the individual data point
• μ is the population mean
• σ is the population standard deviation. This transforms any normal distribution into a standard normal distribution for easier analysis.
8. Can the standard normal distribution be used for all probability calculations?
No, the standard normal distribution is specifically applicable to data that follows a normal distribution. For other probability distributions, different methods and tables are needed.
9. How do you convert a normal score to a standard normal z-score?
You use the z-score formula: z = (x - μ) / σ. Substitute the raw score (x), population mean (μ), and population standard deviation (σ) into the formula to obtain the standardized z-score.
10. What is the role of symmetry in the standard normal curve?
The symmetry of the standard normal curve simplifies probability calculations. Since the mean, median, and mode are all equal and located at the center (z = 0), probabilities are easily determined using the z-table and taking advantage of the symmetrical nature of the curve.
11. How has calculator technology changed solving standard normal problems?
Calculators and statistical software significantly simplify standard normal distribution calculations. Many now include built-in functions to calculate z-scores and associated probabilities directly, eliminating the need for manual table lookups. This speeds up problem-solving and reduces calculation errors.
12. What is the relationship between the standard normal distribution and the area under the curve?
The total area under the standard normal curve is equal to 1, representing 100% probability. The area under a specific portion of the curve represents the probability of a random value falling within that range. This is frequently determined using z-scores and a z-table.
