the empirical rule applies to distributions that are: Does the empirical rule apply to all distributions?


Around 99.7% of values are within 3 standard deviations of the mean. Around 95% of values are within 2 standard deviations of the mean. In a normal distribution, data are symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center. Every normal distribution is a version of the standard normal distribution that’s been stretched or squeezed and moved horizontally right or left.


Distributions are shown us the nature of random data, it shows us how the data is spread in the given minimum and maximum value of a random variable called range. Data Distributions are nothing but the statical functions that give us the important intuitions about the data and give the various values that are very useful for making a crucial decision about the data. In other words, we can say that the probability distribution is a statical function that gives us all possible information and values that a random variable can consist of. Which means 68 % of the data points belonging to the random variable X fall within the range of the first standard deviation.

Sample questions

For small samples, the assumption of normality is important because the sampling distribution of the mean isn’t known. For accurate results, you have to be sure that the population is normally distributed before you can use parametric tests with small samples. You can use parametric tests for large samples from populations with any kind of distribution as long as other important assumptions are met. Around 99.7% of scores are between 700 and 1,600, 3 standard deviations above and below the mean. Around 95% of scores are between 850 and 1,450, 2 standard deviations above and below the mean. Around 99.7% of values are within 3 standard deviations from the mean.

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\nYou can use the empirical rule only if the distribution of the population is normal. As mentioned above, the empirical rule is beneficial for forecasting outcomes within a data set. When the standard deviation has been determined, the data set can easily be subjected to the empirical rule, which shows where the pieces of data lie within the distribution. When you reasonably expect your data to approximate a normal distribution, the mean and standard deviation become even more valuable, thanks to the empirical rule.

Examples of the Empirical Rule

Is an unbiased estimator of the variance of the population distribution, for any distribution of X that has a finite variance. The mean of the empirical distribution is an unbiased estimator of the mean of the population distribution. Draw out a normal curve with a line down the middle and three to either side. My confidence interval is based on an exact binomial calculation; you might get slightly different answers if you use the normal approximation. This website is using a security service to protect itself from online attacks.

  • To this end, 68% of the observed data will occur within the first standard deviation, 95% will take place in the second deviation, and 97.5% within the third standard deviation.
  • On your graph of the probability density function, the probability is the shaded area under the curve that lies to the right of where your SAT scores equal 1380.
  • 95% of data falls within 2 standard deviations from the mean – between μ – 2σ and μ + 2σ .
  • That is the standard deviation between the three primary percentages of the normal distribution, within which the majority of the data in the set should fall, excluding a minor percentage for outliers.
  • Around 99.7% of values are within 3 standard deviations of the mean.

For stock returns, the deviation is often called volatility. If returns are normally distributed, more than 99 percent of the returns are expected to fall within the deviations of the mean value. Such characteristics of the bell-shaped normal distribution allow analysts and investors to make statistical inferences about the expected return and risk of stocks.

95-99.7 % Rule or Empirical Rule:

The _______ is preferred as a measure of the center of a distribution when the data is strongly skewed or has outliers. Statement says the same thing as statement because \(75\%\) of \(251\) is \(188.25\), so the minimum whole number of observations in this interval is \(189\). Approximately \(99.7\%\) of the IQ scores in the population lie between \(70\) and \(130\). To use the Empirical Rule and Chebyshev’s Theorem to draw conclusions about a data set.


A normal distribution can be described by the empirical rule. The standard deviation of a normal distribution is generally between three and five standard deviations. The three-sigma rule is also called the bell curve rule, since it refers to a data distribution that is within three standard deviations of the average. According to the Empirical Rule, almost all data for a normal distribution is within three standard deviations of the mean. 68% of the data falls within a standard deviation under this rule.

Introduction to Statistics Course

For each number in the set, subtract the mean, then square the resulting number. The median is the value of the spread between the highest and lowest numbers within the set.

  • The majority of newborns have normal birthweight whereas only a few percent of newborns have a weight higher or lower than normal.
  • 68% of the data falls within a standard deviation under this rule.
  • It allows statisticians – or those studying the data – to gain insight into where the data will fall, once all is available.

Since again a fraction of an observation is impossible, \(x\; \). The interval in question is the interval from \(66.8\) inches to \(72.4\) inches. By Chebyshev’s Theorem at most 1∕9 of the scores can be below 62, so the rumor is impossible.

Real Life Examples Of Normal Distribution

And now if we want to gather information about the heights of people then we will use the probability distribution. According to this rule, 68% of the data falls within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations from the mean. In this scenario of increasing competition, most parents, as well as children, want to analyze the Intelligent Quotient level.

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Let’s assume a pharmacy would like to anticipate the number of flu cases, within one standard deviation of the mean, so they can properly stock the shelves with medicine. The pharmacy has determined that the historical mean for the flu during the flu season is 9, with a standard deviation of 3. 99.7% of the observations will NOT fall within two standard deviations of the mean. Below is an illustration of a bell curve or normal distribution.

relative frequency histogram

You don’t know that the sample is from a normal distribution. The two-standard-deviation part of the rule often isn’t too far out with unimodal distributions – but can often be a good deal further away than you got. Statement , which is definitely correct, states that at most \(25\%\) of the time either fewer than \(675\) or more than \(775\) vehicles passed through the intersection. Statement says that half of that \(25\%\) corresponds to days of light traffic. This would be correct if the relative frequency histogram of the data were known to be symmetric.

Suppose we want to know what percentage of the the empirical rule applies to distributions that are falls between the values 99 and 105 in this distribution. The pnorm() function in R returns the value of the cumulative density function of the normal distribution. This means if we draw the Probability Density Function of normal distribution then the Pdf of both sides of the mean value will be the mirror image of each other. The Pdf of the Gaussian distribution is a bell-shaped curve that is symmetric. 99.7% of the data is within 3 standard deviations (σ) of the mean (μ). 95% of the data is within 2 standard deviations (σ) of the mean (μ).