Question

Find the \(z\) -values corresponding to the following percentiles of the standard normal distribution. (a) the 75 th percentile (b) the 90 th percentile (c) the 95 th percentile (d) the 99 th percentile

Short answer

The z-values for the 75th, 90th, 95th, and 99th percentiles of the standard normal distribution are approximately 0.67, 1.28, 1.65, and 2.33 respectively.

Step-by-step solution

Understanding Percentiles in the Normal Distribution

Percentiles in a standard normal distribution represent the values below which a certain percentage of the data falls. To find the z-values for given percentiles, we use a z-table or statistical software that provides the z-values corresponding to the area to the left of the z-value.

Finding the Z-Value for the 75th Percentile

To find the z-value for the 75th percentile, look up the value in the z-table that corresponds to an area of 0.7500 to the left of the z-value, or use statistical software to find this z-value.

Finding the Z-Value for the 90th Percentile

To find the z-value for the 90th percentile, look for the value in the z-table with an area of 0.9000 to the left of the z-value, or use statistical software to determine this z-value.

Finding the Z-Value for the 95th Percentile

The z-value for the 95th percentile corresponds to an area of 0.9500 to the left of the z-value. Again, use the z-table or statistical software to find the exact z-value.

Finding the Z-Value for the 99th Percentile

For the 99th percentile, find the z-value that has an area of 0.9900 to the left of it by consulting the z-table or using statistical software.

Key concepts

Z-Values

When it comes to understanding the standard normal distribution, z-values are a necessitous concept. They are also known as standard scores and reveal how many standard deviations a particular value is from the mean. The beauty of z-values lies in their capacity to standardize different normal distributions, permitting the use of a single reference table to determine probabilities.

Where a myriad of data, each with distinct means and standard deviations, previously seemed chaotic and incommensurable, z-values bring order and comparability. They consist of positive and negative values; whereby, a positive z-value indicates a score above the mean, and a negative one signifies a score below it. In the context of our exercise, finding the z-value corresponding to a percentile involves pinpointing where on the normal distribution curve that percentile's data value would land.

Z-Table

A z-table efficiently encapsulates the percentages of data points that fall within certain z-values in a standard normal distribution. In essence, it serves as a decoder ring, translating between z-values and percentile ranks. Using it requires scanning rows and columns to find the closest area to our sought percentile, thereby revealing the associated z-value.

For students, mastery of the z-table is crucial as it allows them to quickly determine critical values for statistical testing and to assess the percentile rank for a given data point without the need for technology. However, small discrepancies in z-tables can occur, so ensuring you're referencing a table that corresponds to the particular criteria of your problem set is vital.

Normal Distribution

The term normal distribution, often known as the bell curve, typifies a highly predictable pattern of data dispersion. It articulates that data tends to cluster around a central value (the mean) with no skew to the left or right. Its predictability and symmetry allow statisticians to make profound inferences about data sets.

Characteristics of The Normal Distribution

In a perfect normal distribution, the mean, median, and mode are equivalent, and the tails of the curve extend infinitely, embodying all potential data points. Any given score's practicality and occurrence can be estimated through its z-value, underscoring the inseparable nature of z-values and the normal distribution.

Statistical Software

For many, particularly in our digital age, statistical software has become a bedrock tool in the analysis of data. It simplifies a user's engagement with data, performing complex computations and visual data representation with a few clicks. In our percentile finding endeavor, it provides us with precise z-values almost instantaneously.

Given its efficiency and accuracy, statistical software can be particularly handy when dealing with large datasets or when a z-table is not readily available. It can not only find z-values for us but also offer a plethora of functionalities, from running simulations to conducting a swath of statistical tests, all while mitigating human error in calculations.

Percentile Rank

The percentile rank represents the proportion of scores in a distribution that a particular score surpasses. For example, if you're in the 75th percentile, you've outperformed 75% of the peer scores. It's a relatable and often-used metric in standardized testing and academic assessment.

Understanding percentile ranks is integral to interpreting standardized test scores, report cards, and other evaluations typically distributed along a bell curve. Whereas actual scores can sometimes seem abstract, percentile ranks offer a concrete sense of where one stands in comparison to a larger group.