The figure above shows that 34.13% of the area is between the mean and +1 or -1SD units, called a z score. Therefore atotal of 68.26% (34.13% x 2) of the test scores fall between +1 and -1 SD.
In a normal curve, the percentage of scores which fall between -1 and +1 standard deviations (SD) is 68%.
In particular, the empirical rule predicts that in normal distributions, 68% of observations fall within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ) of the mean.
In general, about 68% of the area under a normal distribution curve lies within one standard deviation of the mean. That is, if ˉx is the mean and σ is the standard deviation of the distribution, then 68% of the values fall in the range between (ˉx−σ) and (ˉx+σ) .
This 3-part diagram shows the percent of a normal distribution that lies between 1, 2, and 3 standard deviations from the mean: between -1 and 1 you can find approximately 68%; between -2 and 2 is approximately 95%; and between -3 and 3 is approximately 99.7% -- practically everything!
Answer and Explanation:
In a normal distribution, about 95% of the data are within two standard deviations of the mean. This is according to 68-95-99.7 empirical rule of standard score. The rule states that; About 68% of the data is contained within one standard standard of the mean.
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. - 99.7% of the data points will fall within three standard deviations of the mean.
Regardless of what a normal distribution looks like or how big or small the standard deviation is, approximately 68 percent of the observations (or 68 percent of the area under the curve) will always fall within two standard deviations (one above and one below) of the mean.
About 95% of the x values lie between –2σ and +2σ of the mean µ (within two standard deviations of the mean). About 99.7% of the x values lie between –3σ and +3σ of the mean µ (within three standard deviations of the mean).
The empirical rule applies to the normal or bell-shaped distribution. The empirical rule states that approximately 68 percent of data values fall within 1 standard deviation of the mean.
The 95% Rule states that approximately 95% of observations fall within two standard deviations of the mean on a normal distribution.
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.
For example, we know that the area between z = -1.0 and z = 1.0 (i.e. within one standard deviation of the mean) contains 68% of the area under the curve, which can be represented in decimal form at 0.6800 (to change a percentage to a decimal, simply move the decimal point 2 places to the left).
In any normal distribution, 68 percent of the scores fall within one standard deviation of the mean.
Around 68% of scores are within 1 standard deviation of the mean, Around 95% of scores are within 2 standard deviations of the mean, Around 99.7% of scores are within 3 standard deviations of the mean.
Sixty-eight percent of the data is within one standard deviation (σ) of the mean (μ), 95 percent of the data is within two standard deviations (σ) of the mean (μ), and 99.7 percent of the data is within three standard deviations (σ) of the mean (μ).
The 68-95-99.7 Rule
In the Normal distribution with mean µ and standard deviation σ: Approximately 68% of the observations fall within σ of µ. Approximately 95% of the observations fall within 2σ of µ. Approximately 99.7% of the observations fall within 3σ of µ.
About 68 percent of the x values lie between –1σ and +1σ of the mean µ (within one standard deviation of the mean). About 95 percent of the x values lie between –2σ and +2σ of the mean µ (within two standard deviations of the mean).
In our upcoming lesson on the Empirical Rule, you will see that it is worth memorizing that normally distributed data has the characteristics mentioned above: 50% of all data points are above the mean and 50% are below. Apx 68% of all data points are within 1 standard deviation of the mean.
Standard deviation: It defines how much the observations are dispersed from the mean. The rules of normal distributions are: 68% of the observations will fall within +1SD and -1SD. 95% of the observations will fall within +2SD and -2SD. 99.7% of the observation will fall within +3SD and -3SD.
68% of all scores will fall between a Z score of -1.00 and +1.00. 95% of all scores will fall between a Z score of -2.00 and +2.00. 99.7% of all scores will fall between a Z score of -3.00 and +3.00. 50% of all scores lie above/below a Z score of 0.00.
P ( z < − 1.00 ) is found by reading down the z column to the row − 1.0 and then selecting the probability from the column labeled 0.00 to be 0.1587 . The percentage of a normal distribution is greater than a value that is 1 standard deviation below the mean is 84.13%.
About 68% of the x values lie between –1σ and +1σ of the mean µ (within one standard deviation of the mean). About 95% of the x values lie between –2σ and +2σ of the mean µ (within two standard deviations of the mean).
Since 95% of values fall within two standard deviations of the mean according to the 68-95-99.7 Rule, simply add and subtract two standard deviations from the mean in order to obtain the 95% confidence interval.