For this example question the X-value is your SAT score, 1100. Step 1: Write your X-value into the z-score equation. How well did you score on the test compared to the average test taker? The mean score for the SAT is 1026 and the standard deviation is 209. However, if you don’t have either, you can calculate it by hand.Įxample question: You take the SAT and score 1100. You can easily calculate a z-score on a TI-83 calculator or in Excel. Therefore, there’s less than 1% probability that any sample of women will have a mean height of 70″.Ĭonfused about when to use σ and when to use σ √n? See: Sigma / sqrt (n) - why is it used?īack to Top 3. We also know that 99% of values fall within 3 standard deviations from the mean in a normal probability distribution (see 68 95 99.7 rule). The key here is that we’re dealing with a sampling distribution of means, so we know we have to include the standard error in the formula. What is the probability of finding a random sample of 50 women with a mean height of 70″, assuming the heights are normally distributed? This z-score will tell you how many standard errors there are between the sample mean and the population mean.Įxample problem: In general, the mean height of women is 65″ with a standard deviation of 3.5″. When you have multiple samples and want to describe the standard deviation of those sample means ( the standard error), you would use this z score formula: Z Score Formula: Standard Error of the Mean However, the steps for solving it are exactly the same. This is exactly the same formula as z = x – μ / σ, except that x̄ (the sample mean) is used instead of μ (the population mean) and s (the sample standard deviation) is used instead of σ (the population standard deviation). You may also see the z score formula shown to the left. In this example, your score is 1.6 standard deviations above the mean. The z score tells you how many standard deviations from the mean your score is. Assuming a normal distribution, your z score would be: The test has a mean (μ) of 150 and a standard deviation (σ) of 25. The basic z score formula for a sample is:įor example, let’s say you have a test score of 190. Z Score Formulas The Z Score Formula: One Sample A z-score can tell you where that person’s weight is compared to the average population’s mean weight.īack to Top 2. For example, knowing that someone’s weight is 150 pounds might be good information, but if you want to compare it to the “ average” person’s weight, looking at a vast table of data can be overwhelming (especially if some weights are recorded in kilograms). Results from tests or surveys have thousands of possible results and units those results can often seem meaningless. Z-scores are a way to compare results to a “normal” population. In order to use a z-score, you need to know the mean μ and also the population standard deviation σ. Z-scores range from -3 standard deviations (which would fall to the far left of the normal distribution curve) up to +3 standard deviations (which would fall to the far right of the normal distribution curve). But more technically it’s a measure of how many standard deviations below or above the population mean a raw score is.Ī z-score can be placed on a normal distribution curve. 2.) z = (x – μ)⁄ σ z = (11 – 15)⁄ 2 = -4⁄ 2 = -2 3.) The z-score of the bicycle’s weight is -2.Simply put, a z-score (also called a standard score) gives you an idea of how far from the mean a data point is. Solution: 1.) We have all needed values to apply the population z-score formula. If a particular bicycle weighs 11 kilograms, what is its z-score? The population standard deviation is determined to be 2. Z-Score Example Problem It is determined that the average weight of all bicycles is 15 kilograms. The resulting z-score formula for a sample of data is given as: z = (x – x̄)⁄ S To do so, we replace μ with x̄ (the sample mean) and replace σ with S (the sample standard deviation). We can also calculate the z-score for a sample of data. The z-score formula is given as: z = (x – μ)⁄ σ Where z is the z-score, x is the raw score, μ is the population mean, and σ is the population standard deviation. If the raw score is lesser than the mean, the z-score will be negative. If the raw score (the observed data point) is greater than the mean (the average of all data points), the z-score will be positive. In statistics, the z-score (also called the standard score) is the number of standard deviations a raw score is from the mean.
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