Calculating Z-Score For A Normal Distribution With Mean 90 And Standard Deviation 18

In the realm of statistics, understanding data distribution is paramount. One of the most crucial concepts in statistical analysis is the normal distribution, often visualized as a bell curve. This symmetrical distribution is characterized by its mean (average) and standard deviation (a measure of data spread). The z-score, also known as the standard score, is a statistical measure that quantifies the distance of a particular data point from the mean of a distribution, expressed in terms of standard deviations. In simpler terms, the z-score tells you how many standard deviations a data point is away from the average. A positive z-score indicates that the data point is above the mean, while a negative z-score indicates that it is below the mean. A z-score of zero means the data point is exactly at the mean.

The Significance of Z-Scores

Z-scores play a vital role in various statistical analyses. They allow us to standardize data from different distributions, making it possible to compare values that might otherwise be incomparable. For example, if you scored 80 on a math test and 75 on an English test, you can't directly compare these scores because the tests might have different scales and levels of difficulty. However, by converting these scores to z-scores, you can determine your relative performance in each subject compared to the rest of the class. Z-scores are also instrumental in calculating probabilities associated with specific data points in a normal distribution. This is often done using a standard normal distribution table or a calculator with statistical functions. Furthermore, z-scores are used in hypothesis testing to determine whether a sample mean is significantly different from a population mean. By calculating the z-score for the sample mean, you can assess the likelihood of observing such a mean if the null hypothesis (no difference between the sample and population means) is true. This helps in making informed decisions about the validity of research findings.

Calculating Z-Scores: The Formula

The formula for calculating a z-score is relatively straightforward:

z = (X - μ) / σ

Where:

• z is the z-score
• X is the data point (the value you're interested in)
• μ (mu) is the mean of the distribution
• σ (sigma) is the standard deviation of the distribution

This formula essentially subtracts the mean from the data point to find the deviation and then divides by the standard deviation to express this deviation in terms of standard deviations. The result is the z-score, a standardized measure of the data point's position relative to the mean.

Now, let's dive into a practical example. Consider a dataset that follows a normal distribution with a mean (μ) of 90 and a standard deviation (σ) of 18. Our objective is to determine the approximate z-score for a specific value, X = 64. This value represents a data point within our distribution, and we want to understand its relative position compared to the average. By calculating the z-score, we can quantify how many standard deviations away from the mean this data point lies, providing valuable insights into its significance within the dataset.

Step 1: Identify the Given Values

First, we need to clearly identify the values provided in the problem. We have:

• Mean (μ) = 90
• Standard Deviation (σ) = 18
• Data Point (X) = 64

These values are the foundation for our calculation. The mean gives us the central tendency of the data, the standard deviation tells us the spread or variability, and the data point is the specific value we want to analyze.

Step 2: Apply the Z-Score Formula

Next, we plug these values into the z-score formula:

z = (X - μ) / σ z = (64 - 90) / 18

This step involves substituting the known values into the formula. We subtract the mean (90) from the data point (64) to find the difference, and then divide the result by the standard deviation (18). This process transforms the original data point into a standardized score, allowing us to compare it to other data points in the distribution.

Step 3: Calculate the Result

Now, we perform the arithmetic: z = -26 / 18 z ≈ -1.44

This calculation yields a z-score of approximately -1.44. The negative sign indicates that the data point (64) is below the mean (90). The magnitude of the z-score (1.44) tells us how many standard deviations away from the mean the data point is. In this case, the data point is 1.44 standard deviations below the mean.

The calculated z-score of approximately -1.44 provides valuable information about the position of the data point (64) within the distribution. A negative z-score signifies that the data point is below the mean, which we already established. The magnitude of the z-score, 1.44, indicates that the data point is 1.44 standard deviations below the mean. This allows us to understand the relative rarity or commonality of the data point within the dataset.

Understanding the Position Relative to the Mean

In the context of a normal distribution, a z-score of -1.44 suggests that the value 64 is somewhat below average. Recall that a z-score of 0 represents the mean itself. As z-scores move away from 0 in either direction, they indicate values that are increasingly different from the mean. A z-score of -1.44 is far enough from 0 to suggest that the value 64 is not a typical observation in this distribution. To further interpret this z-score, we can refer to a standard normal distribution table or use statistical software. These tools allow us to determine the proportion of data that falls below a given z-score. For a z-score of -1.44, approximately 7.49% of the data falls below this value. This means that the value 64 is in the lower tail of the distribution, indicating it is a relatively low value compared to the rest of the data.

Implications and Contextual Significance

The interpretation of a z-score is highly context-dependent. In some situations, a z-score of -1.44 might be considered quite significant, while in others, it might be within a typical range of variation. For example, if we were analyzing exam scores, a z-score of -1.44 might indicate a student who is struggling compared to their peers. However, if we were analyzing daily temperature variations, a z-score of -1.44 might represent a relatively mild day compared to the average. Therefore, it's crucial to consider the specific context and the nature of the data when interpreting z-scores. Additionally, z-scores can be used to compare data points from different distributions. By standardizing the data, we can make meaningful comparisons even if the original distributions have different means and standard deviations. This is particularly useful in fields like finance, where comparing investment returns or risk levels across different assets is essential.

In summary, the approximate z-score for the value 64 in a normal distribution with a mean of 90 and a standard deviation of 18 is -1.44. This z-score indicates that the value 64 is 1.44 standard deviations below the mean, placing it in the lower tail of the distribution. Understanding z-scores is a fundamental skill in statistics, enabling us to standardize data, compare values, and interpret their significance within a distribution. By mastering the z-score formula and its applications, you gain a powerful tool for data analysis and decision-making.