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In normally distributed data, about 34% of the values lie between the mean and one standard deviation below the mean, and 34% between the mean and one standard deviation above the mean. In addition, 13.5% of the values lie between the first and second standard deviations above the mean.
Note: The average and standard deviation are expressed as percentages, while the variance is a decimal number.
The standard deviation of the normal distribution is about 10, provided that the mean of the data is approximately 100. The normal distribution can take on any value for the mean and the standard deviation, provided that the data appear to be normally distributed.
Empirical Rule or 68-95-99.7% Rule
Approximately 68% of the data fall within one standard deviation of the mean. Approximately 95% of the data fall within two standard deviations of the mean. Approximately 99.7% of the data fall within three standard deviations of the mean.
In a normal distribution, the area between the mean/median (it's the same thing in a symmetric distribution) and +1 standard deviation is about 34.4%. The area between -1 and +1 is about 68%. That means if you pick a random point, there is about a 2/3 probability of it falling between -1 and +1.
The standard normal distribution always has a mean of zero and a standard deviation of one.
If you have 100 items in a data set and the standard deviation is 20, there is a relatively large spread of values away from the mean. If you have 1,000 items in a data set then a standard deviation of 20 is much less significant.
The empirical rule (also called the "68-95-99.7 rule") is a guideline for how data is distributed in a normal distribution. The rule states that (approximately): - 68% of the data points will fall within one standard deviation of the mean. - 95% of the data points will fall within two standard deviations of the mean.
The empirical rule in statistics, also known as the 68 95 99 rule, states that for normal distributions, 68% of observed data points will lie inside one standard deviation of the mean, 95% will fall within two standard deviations, and 99.7% will occur within three standard deviations.
The range rule of thumb formula is the following: Subtract the smallest value in a dataset from the largest and divide the result by four to estimate the standard deviation. In other words, the StDev is roughly ¼ the range of the data.
A result that has a statistical significance of five sigma means the almost certain likelihood that a bump in the data is caused by a new phenomenon, rather than a statistical fluctuation. Scientists calculate this by measuring the signal against the expected fluctuations in the background noise across the whole range.
Statisticians have determined that values no greater than plus or minus 2 SD represent measurements that are are closer to the true value than those that fall in the area greater than ± 2SD. Thus, most QC programs require that corrective action be initiated for data points routinely outside of the ±2SD range.
Figure 1.10: About 95% of the data are within 2 SD of the mean. Most statisticians define likely results as those that are within two standard deviations of the mean. Anything more than two standard deviations from the mean would be called unlikely.
Your textbook uses an abbreviated form of this, known as the 95% Rule, because 95% is the most commonly used interval. The 95% Rule states that approximately 95% of observations fall within two standard deviations of the mean on a normal distribution.
Around 68% of values are within 1 standard deviation of the mean. Around 95% of values are within 2 standard deviations of the mean. Around 99.7% of values are within 3 standard deviations of the mean.
In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively.
Remember that the rule applies to all normal distributions.
The empirical rule, or the 68-95-99.7 rule, tells you where most of the values lie in a normal distribution: Around 68% of values are within 1 standard deviation of the mean. Around 95% of values are within 2 standard deviations of the mean. Around 99.7% of values are within 3 standard deviations of the mean.
The 68-95-99.7 Rule Example
In this example, the population mean is 100 and the standard deviation is 15. Based on the 68-95-99.7 Rule, approximately 68% of the individuals in the population have an IQ between 85 and 115. Values in this particular interval are the most frequent.
The higher the CV, the higher the standard deviation relative to the mean. In general, a CV value greater than 1 is often considered high. For example, suppose a realtor collects data on the price of 100 houses in her city and finds that the mean price is $150,000 and the standard deviation of prices is $12,000.
If you would have expected a greater percentage to fall between 63 and 95, then your standard deviation may be considered large, and if you would have expected a smaller percentage, then your standard deviation may be considered small.
Answer and Explanation:
A standard deviation of 0 means that all the values in the dataset are the same, and thus have no deviation from the average.
When you standardize a normal distribution, the mean becomes 0 and the standard deviation becomes 1. This allows you to easily calculate the probability of certain values occurring in your distribution, or to compare data sets with different means and standard deviations.
If all the values in a distribution are transformed to Z scores, then the distribution will have a mean of 0 and a standard deviation of 1. This process of transforming a distribution to one with a mean of 0 and a standard deviation of 1 is called standardizing the distribution.