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Understanding z-scores is crucial when working with normal distributions. A **z-score** tells us how many standard deviations an element is from the mean.
To calculate it, use the formula: .
In our example, we're trying to find the probability that a male has a back-to-knee length < 21 inches. First, we standardize this value by calculating its z-score.
Here’s the formula:

.

• Mean (μ): 23.5 inches
• Standard Deviation (σ): 1.1 inches
• Measured Value (X): 21 inches

Apply the given values: .Calculate the result step-by-step:

The final z-score is -2.27. This means that 21 inches is 2.27 standard deviations below the mean.

Now that you've calculated the z-score, the next step is to find the corresponding **probability**. This tells us how likely a given event is to occur.
For a normal distribution, we use the **z-score** to find this probability. First, understand that probabilities in a normal distribution curve range from 0 to 1.
If we're looking up the probability for a z-score of -2.27, it means we need the area under the normal distribution curve to the left of this z-score.
This area represents the percentage of values that fall below 21 inches. We can find this using the **Standard Normal Distribution Table* or a z-score table, which is a statistical tool.
Each value in the table corresponds to a particular probability. For any z-score, simply locate it in the table to find the associated probability.
In our case, the table shows that a **z-score** of -2.27 corresponds to a probability of approximately 0.0116.
Therefore, the probability that a male's back-to-knee length is less than 21 inches is approximately **0.0116** or **1.16%**.

The **Standard Normal Distribution Table** is vital for finding probabilities in a normal distribution. This table gives the area (or probability) to the left of a given z-score in a standard normal distribution.
Here's a quick guide on how to use it:

• Find the row corresponding to the first two digits of your z-score
• Locate the column corresponding to the second decimal place

Let's say our z-score is -2.27:

• First two digits: -2.2
• Second decimal place: 0.07

Combining these two finds the probability value in the table: 0.0116.
Keep in mind that this table only provides positive z-scores, because the normal distribution is symmetric. For negative z-scores like -2.27, the probabilities look the same as for positive values, just on the opposite side of the mean.
Use the table by following general rules:

• If your z-score is positive, look it up directly.
• If your z-score is negative, also look up the positive counterpart and infer the result (since tables show cumulative areas).

Being comfortable with this table greatly empowers your understanding of probabilities in a normal distribution.

The **normal distribution graph** is one of the most useful tools for visualizing data. It is characterized by its bell-shaped curve and is symmetrical about the mean.

• Mean (μ): Central peak
• Standard Deviation (σ): Measures the spread or width

For our exercise, the mean back-to-knee length for males is 23.5 inches, and the standard deviation is 1.1.
The graph shows how data is distributed around the mean. The highest point of the bell curve corresponds to the mean. As you move away from the mean, the curve decreases, indicating fewer occurrences of extreme values.
You can visualize the probability we calculated earlier by shading the area of the curve to the left of z = -2.27. This shaded area represents 1.16% of the total area under the curve, depicting the proportion of males having a sitting back-to-knee length less than 21 inches.
Understanding the normal distribution graph helps you grasp the concept of probabilities in normally distributed data. Not all datasets will form a perfect bell curve, but this graph provides a strong foundation for analyzing and interpreting data distributions.