6.4: Formulas for Chapter 6

Statistics C1000: Introduction to Statistics · LibreTexts

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    Chapter 6 Formulas

    Cutoff Score

    The cutoff score in a normal distribution is the value that separates a specified proportion of the data from the rest, based on a given probability. It is used to identify thresholds, such as the top 10% or bottom 5% of values. The formula to find the cutoff score \(x\) is provided directly below.

    \( x = \mu + z \cdot \sigma \)

    Where:

    • \(\mu\) is the mean
    • \(\sigma\) is the standard deviation
    • \(z\) is the z-score corresponding to the desired cumulative probability from the standard normal distribution table

    z-Score for Normal Distribution

    The z-score is used to convert a value from a normal distribution to the standard normal distribution. It measures how many standard deviations a value is from the mean. This allows comparisons across different normal distributions.

    \( z = \dfrac{x - \mu}{\sigma} \)

    Where:

    • \(x\) the raw score or observed value from the original normal distribution.
    • \(\mu\) is the mean
    • \(\sigma\) is the standard deviation

    z-Score for Distribution of Sample Means

    The z-score for the distribution of sample means measures how many standard errors a sample mean \(\bar{x}\) is from the population mean \(\mu\). It is used to standardize a sample mean when the sampling distribution is approximately normal.

    \( z = \dfrac{\bar{x} - \mu}{\dfrac{\sigma}{\sqrt{n}}} \)

    Where:

    • \(\bar{x}\) is the the sample mean
    • \(\mu\) is the population mean
    • \(\sigma\) is the population standard deviation
    • \(n\) is the sample size
    • \(\dfrac{\sigma}{\sqrt{n}}\) is the standard error of the mean (i.e., the standard deviation of the sampling distribution of the sample mean)

    This page titled 6.4: Formulas for Chapter 6 was last modified on Wed, 13 Aug 2025 01:13:53 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Toros Berberyan, Tracy Nguyen, and Alfie Swan.