The units place and the first decimal place are shown in the left hand column, and the second decimal place is displayed across the top row.īut let's get back to the question about the probability that the BMI is less than 30, i.e., P(X 35)? Again we standardize: Note also that the table shows probabilities to two decimal places of Z. In this case, because the mean is zero and the standard deviation is 1, the Z value is the number of standard deviation units away from the mean, and the area is the probability of observing a value less than that particular Z value. This table is organized to provide the area under the curve to the left of or less of a specified value or "Z value". Probabilities of the Standard Normal Distribution Z So, the 50% below the mean plus the 34% above the mean gives us 84%. That is because one standard deviation above and below the mean encompasses about 68% of the area, so one standard deviation above the mean represents half of that of 34%. The table in the frame below shows the probabilities for the standard normal distribution. Examine the table and note that a "Z" score of 0.0 lists a probability of 0.50 or 50%, and a "Z" score of 1, meaning one standard deviation above the mean, lists a probability of 0.8413 or 84%. For any given Z-score we can compute the area under the curve to the left of that Z-score. Since the area under the standard curve = 1, we can begin to more precisely define the probabilities of specific observation. After standarization, the BMI=30 discussed on the previous page is shown below lying 0.16667 units above the mean of 0 on the standard normal distribution on the right. However, when using a standard normal distribution, we will use "Z" to refer to a variable in the context of a standard normal distribution. To this point, we have been using "X" to denote the variable of interest (e.g., X=BMI, X=height, X=weight). For the standard normal distribution, 68% of the observations lie within 1 standard deviation of the mean 95% lie within two standard deviation of the mean and 99.9% lie within 3 standard deviations of the mean. The standard normal distribution is centered at zero and the degree to which a given measurement deviates from the mean is given by the standard deviation. It’s better to use the interquartile range.The standard normal distribution is a normal distribution with a mean of zero and standard deviation of 1. Confidence intervals are often based on the standard normal distribution. It’s a very useful probability distribution and relatively easy to use. When a variable follows a normal distribution, the histogram is bell-shaped and symmetric, and the best measures of central tendency and dispersion are the mean and the standard deviation. The reason why standard deviation is so popular as a measure of dispersion is its relation with the normal distribution which describes many natural phenomena and whose mathematical properties are interesting in the case of large data sets. Standard deviation is equal to 0 if all values are equal (because all values are then equal to the mean).For two data sets with the same mean, the one with the larger standard deviation is the one in which the data is more spread out from the center.A single very extreme value can increase the standard deviation and misrepresent the dispersion. Standard deviation is sensitive to extreme values.Remember the following properties when you are using the standard deviation: This is why, in most situations, it is helpful to assess the size of the standard deviation relative to its mean. For example, a measure of two large companies with a difference of $10,000 in annual revenues is considered pretty close, while the measure of two individuals with a weight difference of 30 kilograms is considered far apart. When you are measuring something that is in the scale of millions, having measures that are close to the mean value doesn’t have the same meaning as when you are measuring something that is in the scale of hundreds. The magnitude of the mean value of the dataset affects the interpretation of its standard deviation. Standard deviation might be difficult to interpret in terms of how large it has to be when considering the data to be widely dispersed. A single extreme value can have a big impact on the standard deviation. However, standard deviation is affected by extreme values. An item selected at random from a data set whose standard deviation is low has a better chance of being close to the mean than an item from a data set whose standard deviation is higher. The data set with the smaller standard deviation has a narrower spread of measurements around the mean and therefore usually has comparatively fewer high or low values. Standard deviation is useful when comparing the spread of two separate data sets that have approximately the same mean.
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