How to find probability with z score

Answer

Finding probability with a z-score involves calculating how many standard deviations a value lies from the mean, then using a standard normal distribution table or software to determine the corresponding probability. The process converts raw data into standardized scores that can be compared across different normal distributions.

What is the formula to find probability using a z-score?

The formula to calculate a z-score is:

$z=\frac{x-\mu }{\sigma }$

In this formula, x represents the raw score, μ represents the population mean, and σ represents the population standard deviation.

After calculating the z-score, you look up the corresponding probability from the standard normal distribution (z) table. The table provides the probability that a value falls below the given z-score. For probabilities greater than the z-score, subtract the table value from 1. For probabilities between two z-scores, subtract the smaller z-score's table value from the larger one.

How do you read a z-table to find probability?

To read a z-table and find the probability value, follow these steps:

1. Identify the z-score whose probability you want to find. Separate the value into two parts: the whole number and first decimal, and the second decimal. For a z-score of 1.25, the first part is 1.2 and the second part is 0.05.
2. Locate the row in the z-table corresponding to the first part (1.2) on the left column.
3. Locate the column corresponding to the second decimal (0.05) on the top row.
4. Find the value at the intersection of this row and column. This value represents the cumulative probability to the left of the z-score. A value of 0.8944 means approximately 89.44% of values fall below a z-score of 1.25.
5. For probabilities to the right of the z-score, subtract the table value from 1. For probabilities between two z-scores, subtract the smaller table value from the larger one.

What are the different types of z-tables?

Yes, there are different types of z-tables, and the method changes depending on which type you use.

Cumulative from left (standard z-table): This table gives the cumulative probability from negative infinity up to the z-score. It represents $P(Z\le z)$, the probability that the variable is less than or equal to the z-score. This is the most common type. To find probabilities greater than the z-score, subtract the table value from 1.

Cumulative from mean: This table shows the probability between the mean $(z=0)$ and the given positive or negative z-score. The table values correspond to the area from the center of the normal curve to the z-score. To find probabilities less than a positive z-score, add 0.5 (the area below the mean) to the table value. For negative z-scores, subtract the table value from 0.5.

Complementary cumulative (right-tail): This table gives the probability from the z-score to positive infinity, representing $P(Z\ge z)$. To find probabilities less than the z-score, subtract the table value from 1.

The key is knowing which table you have. The calculations differ based on the table type, but all reflect areas under the standard normal curve.

How do you calculate probability from z-score step by step?

The following example demonstrates the complete process using adult male heights, which follow a normal distribution with mean $\mu =70$ inches and standard deviation $\sigma =3$ inches.

Problem: Find $P(X<73)$, $P(X>67)$, and $P(67<X<73)$.

Step 1: Calculate z-scores

Convert raw scores to z-scores using the formula:

$z=\frac{x-\mu }{\sigma }$

For x = 73:

$z=\frac{73-70}{3}=1.00$

For x = 67:

$z=\frac{67-70}{3}=-1.00$

Step 2: Read z-table values

Using a standard z-table (cumulative from left): 

Probability Type
Z-Score
Table Value
P(Z ≤ 1.00)
1.00
0.8413
P(Z ≤ -1.00)
-1.00
0.1587

Step 3: Compute probabilities

$P(X<73)=P(Z<1.00)=0.8413$

This means 84.13% of adult males are shorter than 73 inches.

$P(X>67)=P(Z>-1.00)=1-P(Z\le -1.00)=1-0.1587=0.8413$

This means 84.13% of adult males are taller than 67 inches.

$P(67<X<73)=P(-1.00<Z<1.00)=P(Z\le 1.00)-P(Z\le -1.00)=0.8413-0.1587=0.6826$

This means 68.26% of adult males have heights between 67 and 73 inches.

What are common applications of z-score probability?

Z-score probability calculations apply across multiple fields where data follows a normal distribution.

Education and testing

Z-scores assess student performance relative to peers on standardized exams like SAT or IQ tests. A z-score of 1.6 on a college entrance exam indicates the score exceeds 94.52% of test takers, helping identify top performers or students needing additional support.

Healthcare and medicine

Medical professionals monitor patient metrics including newborn weights, blood pressure, and child growth charts. A newborn weighing 7.7 pounds with z = 0.4 falls above 65.54% of peers, supporting early intervention decisions.

Quality control and manufacturing

Factories evaluate product dimensions and defect rates using z-scores. Values exceeding tolerance limits (typically $z>3$ or $z<-3$) flag outliers for quality assurance review.

Finance

Analysts detect unusual stock prices or portfolio returns compared to historical norms. Z-scores identify investment opportunities or risks by quantifying how far current values deviate from expected returns.

Research and hypothesis testing

Researchers estimate event likelihoods and test statistical significance in fields including biology and psychology. Converting data to z-scores enables probability calculations for normally distributed variables.

What are the limitations of finding probability with z-scores?

Several conditions must be met for z-score probability calculations to produce accurate results.

Data distribution requirements

The data must follow a normal (Gaussian) distribution. Z-scores and standard tables apply only to normally distributed populations. Verify normality using tests like Shapiro-Wilk or Q-Q plots before applying z-score methods.

Parameter requirements

Population parameters (mean μ and standard deviation σ) should be known values. Using sample estimates introduces approximation errors that affect probability accuracy.

Sample size considerations

Z-scores can mislead in small datasets $(n<30)$. Extreme values are capped and outliers inflate the mean and standard deviation, causing scores to appear less extreme than they actually are.

Data independence

Non-independent data or unequal variances invalidate z-score results. Avoid using z-scores with highly skewed or multimodal distributions.

Table precision limitations

Standard z-tables provide precision to two decimal places. For higher accuracy, use software functions like Excel's NORM.S.DIST or statistical programming tools.

When these conditions are not met, consider alternatives like t-distributions for small samples or non-parametric methods for non-normal data.

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