Calculating Z-Score For X = 1 In Normal Distribution N(12, 3)

In the realm of statistics, understanding and applying z-scores is crucial for data analysis and interpretation. A z-score, also known as a standard score, quantifies the number of standard deviations a particular data point deviates from the mean of its dataset. This standardized measure allows us to compare data points from different normal distributions, making it a powerful tool in various fields, including finance, healthcare, and engineering. In this comprehensive guide, we will delve into the concept of z-scores, explore their significance, and walk through the step-by-step calculation of a z-score for a specific data point within a normally distributed dataset. We will specifically address the question: What is the z-score of x, when x = 1 and X ~ N(12, 3)? By the end of this article, you will have a solid grasp of z-scores and their practical applications.

The importance of understanding z-scores cannot be overstated. These scores provide a standardized way to measure how far a data point is from the average value in a distribution. In a normal distribution, z-scores help us determine the probability of observing a particular data point, which is essential for hypothesis testing and statistical inference. Furthermore, z-scores enable us to compare data from different datasets that may have different means and standard deviations. This comparative ability is particularly useful in fields where data standardization is necessary for accurate analysis and decision-making. For instance, in finance, z-scores can be used to assess the risk associated with an investment portfolio by comparing its performance against a benchmark index. In healthcare, z-scores can help identify patients with unusual health metrics relative to a normal population, aiding in early diagnosis and treatment. Therefore, mastering the concept of z-scores is a fundamental skill for anyone working with quantitative data.

This article will not only explain the theoretical underpinnings of z-scores but also provide a practical example to illustrate their calculation. We will start by defining the z-score formula and its components, including the data point, the mean, and the standard deviation. Then, we will apply this formula to the given problem: finding the z-score for x = 1 in a normal distribution with a mean of 12 and a standard deviation derived from the variance of 3. Through this example, we will demonstrate how to correctly substitute the values into the formula and compute the z-score. Additionally, we will discuss the interpretation of the calculated z-score, explaining what it means for the data point in relation to the distribution. This hands-on approach will solidify your understanding of z-scores and empower you to calculate them in various scenarios. By the end of this discussion, you will be well-equipped to use z-scores as a powerful tool in your statistical analysis toolkit.

Understanding Z-Scores

The z-score, or standard score, is a dimensionless quantity that indicates the distance between a data point and the mean of its distribution, measured in terms of standard deviations. The z-score is a fundamental concept in statistics, especially when dealing with normal distributions. A normal distribution, often referred to as a bell curve, is symmetric and characterized by its mean (μ) and standard deviation (σ). The mean represents the average value of the dataset, while the standard deviation measures the spread or dispersion of the data around the mean. The z-score allows us to standardize data from different normal distributions, making it possible to compare and analyze them on a common scale. This standardization is crucial for various statistical analyses, including hypothesis testing, confidence interval estimation, and probability calculations.

The formula for calculating the z-score is straightforward yet powerful. It is defined as:

z = (x - μ) / σ

Where:

• z is the z-score.
• x is the data point.
• μ (mu) is the mean of the distribution.
• σ (sigma) is the standard deviation of the distribution.

This formula subtracts the mean from the data point (x - μ), giving the deviation of the data point from the mean. This deviation is then divided by the standard deviation (σ), which scales the deviation into units of standard deviations. The resulting z-score represents how many standard deviations the data point is away from the mean. 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 0 means the data point is exactly at the mean.

Interpreting z-scores is essential for understanding their significance. A z-score of 1, for example, means that the data point is one standard deviation above the mean. In a standard normal distribution (a normal distribution with a mean of 0 and a standard deviation of 1), approximately 68% of the data falls within one standard deviation of the mean (between z = -1 and z = 1), about 95% falls within two standard deviations (between z = -2 and z = 2), and about 99.7% falls within three standard deviations (between z = -3 and z = 3). These percentages are derived from the empirical rule (also known as the 68-95-99.7 rule) and provide a quick way to assess the rarity of a data point. For instance, a z-score of 2 indicates that the data point is relatively unusual, as it falls outside the central 95% of the data. A z-score of 3 or higher is considered extremely rare, suggesting that the data point is an outlier. Therefore, understanding how to interpret z-scores allows us to make informed judgments about the position and significance of data points within a distribution.

Calculating the Z-Score for x = 1

To calculate the z-score for x = 1 when X ~ N(12, 3), we need to apply the z-score formula, which is z = (x - μ) / σ. In this scenario, we are given the data point x = 1, and the distribution X follows a normal distribution N(12, 3). The notation N(12, 3) provides us with two key pieces of information: the mean (μ) and the variance (σ²). The first number, 12, represents the mean (μ) of the distribution. The second number, 3, represents the variance (σ²), not the standard deviation (σ). It is crucial to remember that the z-score formula requires the standard deviation, not the variance. Therefore, the first step in calculating the z-score is to find the standard deviation by taking the square root of the variance.

The standard deviation (σ) is the square root of the variance (σ²). In this case, the variance is given as 3. Thus, we calculate the standard deviation as follows:

σ = √σ² = √3 ≈ 1.732

Now that we have the standard deviation (σ ≈ 1.732), we can proceed with calculating the z-score. We have all the necessary components: the data point x = 1, the mean μ = 12, and the standard deviation σ ≈ 1.732. We will now substitute these values into the z-score formula:

z = (x - μ) / σ

z = (1 - 12) / 1.732

z = -11 / 1.732

z ≈ -6.351

Therefore, the calculated z-score for x = 1 in the normal distribution N(12, 3) is approximately -6.351. This z-score indicates that the data point x = 1 is significantly below the mean of the distribution. To further understand the implications of this z-score, we need to interpret its value within the context of the normal distribution.

Interpretation of the Z-Score

The z-score of approximately -6.351 calculated for x = 1 in the normal distribution N(12, 3) is a crucial piece of information that provides significant insight into the position of this data point within the distribution. A z-score of -6.351 indicates that the data point x = 1 is 6.351 standard deviations below the mean (μ = 12). In the context of a normal distribution, this is an exceptionally low value. To fully appreciate the rarity of this data point, it is essential to understand how z-scores relate to the spread of data in a normal distribution.

In a standard normal distribution, which has a mean of 0 and a standard deviation of 1, approximately 68% of the data falls within one standard deviation of the mean (between z = -1 and z = 1), about 95% falls within two standard deviations (between z = -2 and z = 2), and about 99.7% falls within three standard deviations (between z = -3 and z = 3). This is known as the empirical rule or the 68-95-99.7 rule. Beyond three standard deviations from the mean, the occurrence of data points becomes extremely rare. Our calculated z-score of -6.351 is far beyond three standard deviations below the mean, indicating that the data point x = 1 is a significant outlier in this distribution.

The practical implication of a z-score of -6.351 is that observing a value as low as 1 in a normal distribution with a mean of 12 and a standard deviation of approximately 1.732 is highly improbable. This rarity suggests that either the data point is an anomaly, or the assumption that the data follows a normal distribution with the given parameters may be incorrect. In real-world scenarios, such a low z-score might prompt further investigation to understand the underlying reasons for this deviation. For example, if this data point represents a measurement in a manufacturing process, a z-score of -6.351 might indicate a malfunction in the process. Alternatively, if the data represents test scores, it might suggest an unusual performance. In any case, a z-score of this magnitude serves as a strong signal that the data point is not typical and warrants additional scrutiny. Thus, the interpretation of the z-score is not merely a numerical exercise but a critical step in drawing meaningful conclusions from data analysis.

Conclusion

In conclusion, understanding and calculating z-scores is a fundamental skill in statistics, providing a standardized way to measure how far a data point is from the mean of its distribution. The z-score calculated for x = 1 in the normal distribution N(12, 3) is approximately -6.351. This result signifies that the data point x = 1 is exceptionally far below the mean of 12, specifically 6.351 standard deviations away. Such a low z-score indicates a rare occurrence within this distribution, suggesting that the data point is a significant outlier. The process of calculating the z-score involves using the formula z = (x - μ) / σ, where x is the data point, μ is the mean, and σ is the standard deviation. In this case, we first calculated the standard deviation by taking the square root of the variance (σ = √3 ≈ 1.732) and then applied the formula to find the z-score.

The interpretation of z-scores is crucial for understanding their implications. A z-score of -6.351 falls far outside the typical range of data points in a normal distribution, as defined by the empirical rule (68-95-99.7 rule). This rule tells us that approximately 99.7% of the data lies within three standard deviations of the mean. Therefore, a z-score of -6.351 is well beyond this range, indicating that the data point is highly unusual. In practical applications, such a low z-score might prompt further investigation to determine the cause of the deviation, whether it is due to a measurement error, a process malfunction, or some other underlying factor.

The ability to calculate and interpret z-scores is invaluable in various fields, including finance, healthcare, and engineering. Z-scores enable comparisons between data points from different distributions and provide insights into the probability of observing particular values. By mastering the concept of z-scores, individuals can make more informed decisions and draw meaningful conclusions from data analysis. This guide has provided a comprehensive overview of z-scores, their calculation, and their interpretation, equipping readers with the knowledge and skills necessary to apply this powerful statistical tool in their own work and studies. Understanding z-scores not only enhances statistical literacy but also fosters critical thinking and data-driven decision-making.