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Percentiles are a way to understand how a score compares to others in a set. They indicate the percentage of scores that fall below a particular value. In a normal distribution, different percentiles have typical positions.
Here are some key facts about percentiles:

• The 16th percentile means that 16% of all scores are below this value.
• Percentiles are often used in standardized testing to compare a student's performance to their peers.
• The 84th percentile means that 84% of all scores are below this one, making it a higher score than most. Because the normal curve is symmetric, if the 16th percentile is a certain distance below the mean, the 84th percentile is the same distance above the mean.

When we know the 16th percentile score and the mean, we can calculate the standard deviation, which helps us find other percentiles. For example, in the given exam problem, with a mean score of 100, the 16th percentile is at 80, allowing us to find the 84th percentile at 120.
Understanding percentiles helps learners position their performance and expectations relative to others, especially in normally distributed data.

Standard deviation is a crucial concept in statistics that measures the spread of a set of values around the mean. It tells us how much variation or dispersion there is from the average. In a normal distribution, most values lie within a range defined by the standard deviation. Here are some important points:

• One standard deviation from the mean in each direction covers about 68% of the data.
• Two standard deviations include about 95%, and three standard deviations nearly 99.7%.
• A small standard deviation means values are close to the mean; a large one indicates values are spread out over a wider range.

In our example, the standard deviation is 20. This tells us that the scores vary, on average, by 20 points above or below the mean of 100. Using standard deviation helps us identify outliers and understand the distribution of scores. It's essential for interpreting data variability and making informed decisions in contexts like education and finance.

A z-score, or standard score, tells us how many standard deviations a data point is from the mean. It is a helpful tool for comparing individual scores to a group average, especially in the context of a normal distribution. To calculate a z-score, use the formula:

• \[z = \frac{x - \mu}{\sigma}\]
• where:
  • \[x\] is the individual score,
  • \[\mu\] is the mean, and
  • \[\sigma\] is the standard deviation.

For example, if we want to find the z-score for a score of 90:

• Subtract the mean (100) from 90, which gives -10.
• Divide by the standard deviation (20) to get a z-score of -0.5.

This z-score of -0.5 indicates that a score of 90 is half a standard deviation below the mean. Z-scores allow easy comparison across different datasets by standardizing scores, making them useful in educational assessments, finance, and psychology to gauge individual performance relative to a broader set of data.