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Q: What does it mean to find a data point within one standard deviation of the mean?

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Assuming a normal distribution 68 % of the data samples will be with 1 standard deviation of the mean.

Standard deviation is a measure of variation from the mean of a data set. 1 standard deviation from the mean (which is usually + and - from mean) contains 68% of the data.

The standard deviation is the square root of the variance.

Standard deviation shows how much variation there is from the "average" (mean). A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

The purpose of obtaining the standard deviation is to measure the dispersion data has from the mean. Data sets can be widely dispersed, or narrowly dispersed. The standard deviation measures the degree of dispersion. Each standard deviation has a percentage probability that a single datum will fall within that distance from the mean. One standard deviation of a normal distribution contains 66.67% of all data in a particular data set. Therefore, any single datum in the data has a 66.67% chance of falling within one standard deviation from the mean. 95% of all data in the data set will fall within two standard deviations of the mean. So, how does this help us in the real world? Well, I will use the world of finance/investments to illustrate real world application. In finance, we use the standard deviation and variance to measure risk of a particular investment. Assume the mean is 15%. That would indicate that we expect to earn a 15% return on an investment. However, we never earn what we expect, so we use the standard deviation to measure the likelihood the expected return will fall away from that expected return (or mean). If the standard deviation is 2%, we have a 66.67% chance the return will actually be between 13% and 17%. We expect a 95% chance that the return on the investment will yield an 11% to 19% return. The larger the standard deviation, the greater the risk involved with a particular investment. That is a real world example of how we use the standard deviation to measure risk, and expected return on an investment.

Related questions

Standard deviation is a measure of the spread of data.

The standard deviation is a measure of the spread of data.

The Empirical Rule states that 68% of the data falls within 1 standard deviation from the mean. Since 1000 data values are given, take .68*1000 and you have 680 values are within 1 standard deviation from the mean.

No, if the standard deviation is small the data is less dispersed.

Assuming a normal distribution 68 % of the data samples will be with 1 standard deviation of the mean.

It's used in determining how far from the standard (average) a certain item or data point happen to be. (Ie, one standard deviation; two standard deviations, etc.)

Standard deviation is the variance from the mean of the data.

Standard deviation is a measure of variation from the mean of a data set. 1 standard deviation from the mean (which is usually + and - from mean) contains 68% of the data.

One standard deviation for one side will be 34% of data. So within 1 std. dev. to both sides will be 68% (approximately) .the data falls outside 1 standard deviation of the mean will be 1.00 - 0.68 = 0.32 (32 %)

No. Variance and standard deviation are dependent on, but calculated irrespective of the data. You do, of course, have to have some variation, otherwise, the variance and standard deviation will be zero.

Standard deviation has the same unit as the data set unit.

I am not entirely sure I understand correctly what you mean by "essence". However, the idea of finding the standard deviation is to determine, as a general tendency, whether most data points are close to the average, or whether there is a large spread in the data. The standard deviation means, more or less, "How far is the typical data point from the average?"

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