Analyzing Z-scores To Determine Dataset Characteristics

When analyzing data, understanding the distribution and key statistical measures is crucial. Data analysis often involves interpreting individual data points in relation to the entire dataset. In this article, we will explore how z-scores and the provided information about $z_{20} = -2$ and $z_{50} = -1$ can help us infer properties of the dataset, such as the mean, standard deviation, and the relative position of data points within the distribution. We will delve into each of the given options (variance, standard deviation, mean, median, and the significance of the data point x = 20) and discuss which conclusions can be reliably drawn from the available data. This analysis will provide a comprehensive understanding of how statistical measures interact and how they help in making informed decisions based on data.

Understanding Z-Scores

To effectively analyze the given problem, it's essential to first grasp the concept of z-scores. A z-score, also known as a standard score, indicates how many standard deviations a data point is from the mean of the dataset. The formula for calculating a z-score is:

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

Where:

• $z$ is the z-score
• $x$ is the data point
• $\mu$ is the mean of the dataset
• $\sigma$ is the standard deviation of the dataset

A positive z-score indicates that the data point is above the mean, while a negative z-score indicates that the data point is below the mean. The magnitude of the z-score represents the distance from the mean in terms of standard deviations. For instance, a z-score of 2 means the data point is 2 standard deviations above the mean, and a z-score of -1.5 means the data point is 1.5 standard deviations below the mean.

In the given problem, we have two data points and their corresponding z-scores: $z_{20} = -2$ and $z_{50} = -1$. This means that the data point 20 is 2 standard deviations below the mean, and the data point 50 is 1 standard deviation below the mean. By using these two pieces of information, we can set up a system of equations to solve for the mean ($\mu$) and the standard deviation ($\sigma$) of the dataset. This initial setup is crucial for evaluating the validity of the options presented.

Setting Up the Equations

Given the z-score formula and the information provided ($z_{20} = -2$ and $z_{50} = -1$), we can set up two equations:

1. For $x = 20$ and $z = -2$: $-2 = \frac{20 - \mu}{\sigma}$ This can be rewritten as: $-2\sigma = 20 - \mu$ Or: $\mu - 2\sigma = 20$ (Equation 1)
2. For $x = 50$ and $z = -1$: $-1 = \frac{50 - \mu}{\sigma}$ This can be rewritten as: $-\sigma = 50 - \mu$ Or: $\mu - \sigma = 50$ (Equation 2)

We now have a system of two linear equations with two variables ($\mu$ and $\sigma$). This system can be solved using various methods, such as substitution or elimination. The solution to this system will give us the values for the mean and the standard deviation, which are essential for analyzing the given options.

Solving for Mean and Standard Deviation

To solve the system of equations, we can use the method of substitution or elimination. Here, we will use the elimination method.

We have the two equations:

1. $\mu - 2\sigma = 20$
2. $\mu - \sigma = 50$

Subtract Equation 1 from Equation 2:

$(\mu - \sigma) - (\mu - 2\sigma) = 50 - 20$

This simplifies to:

$\sigma = 30$

Now that we have the value of $\sigma$, we can substitute it into either Equation 1 or Equation 2 to solve for $\mu$. Let's use Equation 2:

$\mu - 30 = 50$

Adding 30 to both sides, we get:

$\mu = 80$

Thus, the standard deviation ($\sigma$) is 30, and the mean ($\mu$) is 80. These values are crucial for evaluating the options provided in the question.

Evaluating the Options

Now that we have determined the mean ($\mu = 80$) and the standard deviation ($\sigma = 30$), we can evaluate each of the given options:

A. The variance is 10.

The variance is the square of the standard deviation. In this case:

$Variance = \sigma^2 = 30^2 = 900$

So, this option is incorrect. The variance is 900, not 10.

B. The standard deviation is 30.

As we calculated, the standard deviation is indeed 30. So, this option is correct.

C. The mean is 80.

We calculated the mean to be 80. So, this option is also correct.

D. The median is 40.

We do not have enough information to determine the median. Knowing the mean and standard deviation does not directly tell us the median, especially without knowing the shape of the distribution. The median could be equal to the mean in a symmetrical distribution, but without that knowledge, we cannot confirm this option.

E. The data point x = 20 is 2 standard deviationsDiscussion category.

This statement refers to the z-score of the data point 20. We were given that $z_{20} = -2$, which means 20 is 2 standard deviations below the mean. The statement lacks the crucial detail of direction (below the mean), thus making it misleading.

Final Answer

Based on our analysis, the correct options are:

• B. The standard deviation is 30.
• C. The mean is 80.

These conclusions are drawn directly from the calculated values of the standard deviation and the mean using the z-score formula and the given data points. Understanding how to use z-scores to derive key statistical measures is fundamental in data analysis and interpretation. This problem exemplifies how two data points and their z-scores can provide significant insights into the distribution of a dataset.