91Ó°ÊÓ

When dealing with the normal distribution, a useful measure called the z-score helps determine how far a specific data point is from the mean. Basically, it's a numerical representation that tells you how many standard deviations a particular value is away from the mean of a dataset.
To calculate a z-score, use the formula:

• \( z = \frac{x - \mu}{\sigma} \)

Where:

• \( x \) represents the data point you're analyzing.
• \( \mu \) is the mean of the dataset.
• \( \sigma \) is the standard deviation of the dataset.

A positive z-score indicates the data point is above the mean, while a negative one signifies it is below the mean. This measure allows us to use standardized tables or calculators to find probabilities and percentages associated with a certain value in a normal distribution.
For example, in our exercise, we calculated a z-score of approximately 1.28, indicating that the required test time is 1.28 standard deviations above the mean, making it possible to predict that 90% of students will complete the test within that timeframe.

Standard deviation is a key concept in statistics, often used to describe the amount of variation or dispersion in a set of values. It effectively measures how spread out the data points are from the mean.
The formula for standard deviation, \( \sigma \), involves a series of steps:

• Calculating the mean (average) of the dataset.
• Subtracting the mean from each data point to find the deviations.
• Squaring each deviation to make all values positive.
• Finding the average of these squared deviations.
• Taking the square root of this average to obtain the standard deviation.

In our exercise, the standard deviation was provided as 12 minutes, indicating the typical deviation from the average test completion time of 70 minutes. A smaller standard deviation would suggest that most students finish close to this mean time, whereas a larger one would indicate a broader spread of completion times across students.

The mean, often known as the average, is one of the most commonly used measures in statistics to summarize a set of data. It provides a central value around which the data clusters. To calculate the mean, you sum up all the data points in a set and then divide this total by the number of data points.

• If you have a set of numbers: \( x_1, x_2, \ldots, x_n \)
• The mean \( \mu \) can be calculated using:
• \( \mu = \frac{x_1 + x_2 + \cdots + x_n}{n} \)

In the context of our exercise, the mean test completion time was found to be 70 minutes. This figure tells us that on average, most students finish the achievement test within this timeframe. Understanding the mean helps gauge overall performance and set realistic time limits for events such as exams based on average completion rates.