Want to calculate a z-score for a raw score or sample mean? Use the Z-score calculator.
How to Use the Z Score Probability Calculator
Want to convert a z-score to probability? This calculator helps you quickly find the area to the left of z, area to the right of z, area between 0 and a z-score, area between two z-scores, or area outside two different z-scores. To find the correct probability from z using this tool, follow these steps:
- Choose the probability type that matches your question. Specifically, you should choose:
- P(Z < z) to find the area to the left of a z-score
- P(Z > z) to find the area to the right of a z-score
- P(0 to z) to find the area between 0 and a z-score
- P(z₁ < Z < z₂) to find the area between two z scores
- Outside z₁ and z₂ to find the area in both tails outside two z scores
- Enter the z score value (s)
- Click Calculate
After you click Calculate, the calculator shows the probability, interpretation, and a graph showing the area under the normal curve. It also provides a clear, step-by-step solution, explaining how to find the probability from the z table. This helps you understand where the answer comes from instead of only copying the final result.
Note. You can enter positive or negative z scores. For between and outside calculations, enter both z scores. If the values are not in order, the calculator will still use them as the lower and upper bounds when finding the area.
.Example 1: Area to the Left of a Z Score
A statistics exam has a mean score of 70 and a standard deviation of 10. A student scored 85. What percentage of students scored less than 85?
Solution
Step 1: Identify the values
From the question:
- Raw score: x = 85
- Mean: μ = 70
- Standard deviation: σ = 10
Since the question asks for the percentage of students who scored less than 85, we need the area to the left of the z-score. In other words, we need to find P(X < 85).
Step 2: Convert the raw score to a z score
By definition, z = (x − μ) / σ
Substituting the values into the formula gives: z = (85 − 70) / 10
= 1.50
Step 3: Use the calculator to find the probability
Since we’ve standardized x to z, we need to find P(Z< 1.50). Using the calculator:
- Select P(Z<z)
- Enter 1.50 as the z score
- Click Calculate
The calculator returns P(Z < 1.50) = 0.9332 and shows a graph representing this area under the standard normal curve. Therefore, about 93.32% of students scored less than 85.
Example 2: Area to the Right of a Z Score
The weights of a certain product are normally distributed with a mean of 50 grams and a standard deviation of 4 grams. What is the probability that a randomly selected product weighs more than 56 grams?
Solution
Step 1: Identify the values
From the question:
- Raw value: x = 56
- Mean: μ = 50
- Standard deviation: σ = 4
Since the question asks for the probability of a product weighing more than 56 grams, we need the area to the right of the z-score. That’s, P(X> 56)
Step 2: Convert the raw value to a z score
By definition, z = (x − μ) / σ
Substituting the values into the formula and solving gives:
z = (56 − 50) / 4
= 1.50
Step 3: Use the calculator to find the probability
Since we’ve standardized the x value to z, we need to find P(Z > 1.50). To do this using the calculator:
- Choose P(Z > z)
- Enter 1.50 as the z score
- Click calculate
The calculator gives P(Z > 1.50) = 0.0668 and displays a graph showing this area under the standard normal curve. Therefore, the probability that a randomly selected product weighs more than 56 grams is 0.0668, or 6.68%.
Example 3: Area Between 0 and a Negative Z Score
The daily time spent on a learning app is normally distributed with a mean of 60 minutes and a standard deviation of 12 minutes. A student uses the app for 48 minutes. What proportion of users spend between 48 minutes and the mean time of 60 minutes?
Solution
Step 1: Identify the values
From the question:
- Raw value: x1 = 48 and x2= 60
- Mean: μ = 60
- Standard deviation: σ = 12
Since the question asks for the area between 48 minutes and the mean time of 60 minutes, we need to find P(48 <X< 60)
Step 2: Convert the raw value to a z score
By definition, z = (x − μ) / σ
For x1 = 48, z = (48 − 60) / 12
= -1
However, for x2 = 60
z = (60-60)/12
= 0
This means we need to find P(-1< Z < 0)
Step 3: Use the calculator to find the probability
To find P(-1<Z<0) using the calculator:
- Choose P(0 to z)
- Enter -1 as the Z score
- Click Calculate
The calculator gives: P(-1 < Z < 0) = 0.3413 and displays a graph representing this area under the standard normal curve. This implies that the proportion of users spending between 48 minutes and the mean time of 60 minutes is 0.3413 or 34.13%.
Example 4: Area Between Two Z Scores
The heights of adult men in a city are normally distributed with a mean of 175 cm and a standard deviation of 7 cm. What is the probability that a randomly selected man is between 168 cm and 185 cm tall?
Solution
Step 1: Identify the values
From the question:
- Lower value: x₁ = 168
- Upper value: x₂ = 185
- Mean: μ = 175
- Standard deviation: σ = 7
Since the question asks for the probability that a value falls between 168 cm and 185 cm, we need the area between two z-scores. In other words, we need to find P(168 <X<185).
Step 2: Convert both raw values to z scores
For 168 cm: z₁ = (x₁ − μ) / σ
= (168 − 175) / 7
= – 1.00
For 185 cm: z₂ = (x₂ − μ) / σ
= (185 − 175) / 7
= 1.43
Step 3: Use the calculator to find the probability
After standardizing the x values to z, we need to find P(-1.00 < Z < 1.43). To find this probability using the calculator:
- Choose P(z₁ < Z < z₂)
- Enter -1.00 as the first z score
- Enter 1.43 as the second z score
- Click Calculate
The calculator gives P(-1 < Z < 1.43) = 0.7650 and displays a graph representing this area under the standard normal curve. Therefore, about 76.5% of adult men have heights between 168 cm and 185 cm.
Example 5: Area Outside Two Z Scores
A machine fills bottles with a mean volume of 500 ml and a standard deviation of 5 ml. Bottles are rejected if they contain less than 490 ml or more than 510 ml. What proportion of bottles will be rejected?
Solution
Step 1: Identify the values
From the question:
- Lower cutoff: x₁ = 490
- Upper cutoff: x₂ = 510
- Mean: μ = 500
- Standard deviation: σ = 5
Since bottles are rejected if they are below 490 ml or above 510 ml, we need the area outside two z-scores. In other words, we need to find P(X <490 or X > 510).
Step 2: Convert both cutoff values to z scores
For 490 ml: z₁ = (x₁ − μ) / σ
= (490 − 500) / 5
= -2.0
For 510 ml: z₂ = (x₂ − μ) / σ
= (510 − 500) / 5
= 2
Step 3: Use the calculator to find the probability
After standardizing the x values to z, we need to find P(Z < -2.0 or Z > 2.0). To find this probability using the calculator:
- Choose Outside z₁ and z₂
- Enter -2.00 as the first z score
- Enter 2.00 as the second z score
- Click Calculate
The calculator gives: P(Z < -2.00 or Z > 2.00) = 0.0455 and displays a graph representing this area under the standard normal curve. Therefore, abou 4.55% of bottles will be rejected.
What Is a Z Score Probability?
A z-score probability is the area under the standard normal curve for a given z-score or range of z-scores. The standard normal distribution has a mean of 0 and a standard deviation of 1. A z-score tells you how many standard deviations a value is from the mean.
For example, a z score of 1 means the value is one standard deviation above the mean. A z score of -2 means the value is two standard deviations below the mean.
Once a raw value is converted to a z score, you can use the standard normal curve or z table to find the probability. This probability may represent the area to the left, the area to the right, the area between two values, or the area outside two values.
Normal Distribution and Z Scores
The normal distribution is a bell-shaped, symmetric distribution. Many real-world measurements, such as test scores, heights, weights, and product measurements, are often modeled using a normal or approximately normal distribution.
A normal distribution is described by two main values:
- Mean (μ), which gives the center of the distribution
- Standard deviation (σ), which shows how spread out the values are
To use the z-score probability calculator with a normal distribution word problem, you usually convert the raw value to a z score using the formula: z = (x − μ) / σ. Alternatively, you can quickly calculate z values using the z-score calculator.
After finding the z score, use this calculator to find the required area under the curve.
The 68-95-99.7 Rule
The 68-95-99.7 rule is a quick way to understand probabilities in a normal distribution.
It says that approximately:
- 68% of values fall between z = -1 and z = 1
- 95% of values fall between z = -2 and z = 2
- 99.7% of values fall between z = -3 and z = 3
This rule is only an approximation, but it helps you understand whether a z-score is common or unusual.
For exact probabilities, use the calculator instead of relying only on the rule.
Frequently Asked Questions
This is a free online tool that finds the area under the standard normal curve for a given z score. It can calculate left-tail, right-tail, between, and outside probabilities.
Choose P(Z < z), enter the z score, and click Calculate. The result is the cumulative area to the left of that z score.
Choose P(Z > z) and enter the z score. The calculator finds the right-tail probability by subtracting the left-tail probability from 1.
Choose P(z₁ < Z < z₂) and enter both z scores. The calculator subtracts the cumulative probability of the lower z score from the cumulative probability of the upper z score.
Choose Outside z₁ and z₂ and enter the two z scores. The calculator adds the left-tail area below the lower z score and the right-tail area above the upper z score.
Yes. You can enter positive or negative z scores. Negative z scores are below the mean, while positive z scores are above the mean.
Yes. In the standard normal distribution, probability is represented by the area under the curve. For example, an area of 0.95 means a probability of 95%.
