The normal distribution, also called the z-distribution Example: How frequently do you go to the student store on a 1 to 7 scale? How many days per week is that? Does knowing the standard units of days give us more information? Standard scales convey important information in addition to the value observed. We are now going to learn about a distribution (the z-distribution) that has some very handy attributes and we are going to learn how to translate our observations, scores, distributions, into that scale and back again We say URSA instead of the formal name (University Records System Access) to communicate We translate feet into inches to make addition easier, e.g. We already translate units all the time for convenience. This chapter is about taking advantage of something we know is true to give us an edge in making decisions-we do this by translation With a p-value of less than 0.05, you can conclude that average sleep duration in the COVID-19 lockdown was significantly higher than the pre-lockdown average.2004, S. Since the total area under the curve is 1, you subtract the area under the curve below your z-score from 1.Ī p-value of less than 0.05 or 5% means that the sample significantly differs from the population. To find the p-value to assess whether the sample differs from the population, you calculate the area under the curve above or to the right of your z-score. This means that your sample’s mean sleep duration is higher than about 98.74% of the population’s mean sleep duration pre-lockdown. The table tells you that the area under the curve up to or below your z-score is 0.9874. To find the probability of your sample mean z-score of 2.24 or less occurring, you use the z-table to find the value at the intersection of row 2.2 and column +0.04. FormulaĪ z-score of 2.24 means that your sample mean is 2.24 standard deviations greater than the population mean. To compare sleep duration during and before the lockdown, you convert your lockdown sample mean into a z-score using the pre-lockdown population mean and standard deviation. Then, you find the p-value for your z-score using a z-table.First, you calculate a z-score for the sample mean value.To assess whether your sample mean significantly differs from the pre-lockdown population mean, you perform a z -test: You collect sleep duration data from a sample during a full lockdown.īefore the lockdown, the population mean was 6.5 hours of sleep. Let’s walk through an invented research example to better understand how the standard normal distribution works.Īs a sleep researcher, you’re curious about how sleep habits changed during COVID-19 lockdowns. Step-by-step example of using the z-distribution That means it’s likely that only 6.3% of SAT scores in your sample exceed 1380. Position or shape (relative to standard normal distribution) A small standard deviation results in a narrow curve, while a large standard deviation leads to a wide curve. The standard deviation stretches or squeezes the curve. Increasing the mean moves the curve right, while decreasing it moves the curve left. The mean determines where the curve is centred. In the standard normal distribution, the mean and standard deviation are always fixed.Įvery normal distribution is a version of the standard normal distribution that’s been stretched or squeezed and moved horizontally right or left. However, a normal distribution can take on any value as its mean and standard deviation. Normal distribution vs the standard normal distributionĪll normal distributions, like the standard normal distribution, are unimodal and symmetrically distributed with a bell-shaped curve. You can calculate the standard normal distribution with our calculator below.
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