Is non-negative, and the unit is the same as the measured data. There is a difference between the standard deviation of a population or random variable and the standard deviation of the number of samples in a subset. Standard calculation formula:
Suppose there is a set of values x? ，X？ ，X？ , ... Xn (both real numbers), the average (arithmetic average) is μ, and the formula is shown in figure 1.
Standard deviation is also called standard deviation or experimental standard deviation, and the formula is
Simply put, the standard deviation is a measure of the deviation of the average value of a set of data. The standard deviation is large, indicating that most of the values are quite different from their average values; A smaller standard deviation means that these values are closer to the average.
For example, the average values of {0,5,9, 14} and {5,6,8,9} are all 7, but the standard deviation of the second group is smaller.
Standard deviation can be used to measure uncertainty. For example, in physical science, when repeated measurements are made, the standard deviation of the set of measured values represents the accuracy of these measurements. When determining whether the measured value conforms to the predicted value, the standard deviation of the measured value plays a decisive role: if the measured average value is too far from the predicted value (compared with the standard deviation value at the same time), it is considered that the measured value is contradictory to the predicted value. This is easy to understand, because if the measured value falls outside a certain numerical range, it can be reasonably inferred whether the predicted value is correct or not.
Standard deviation is applied to investment and can be used as an index to measure the stability of return. The greater the standard deviation, the higher the risk, because the income is far from the past average. On the contrary, the smaller the standard deviation, the more stable the income and the less the risk.
For example, six students in Group A and Group B all took the same Chinese exam. Group A scored 95, 85, 75, 65, 55, 45, and Group B scored 73, 72, 765, 438+0, 69, 68, 67. The average score of the two groups is 70, but the standard deviation of group A is about 17.08, and that of group B is about 2. 16, which shows that the gap between students in group A is much larger than that of students in group B. ..
If it is a population (estimated population variance), the root sign is divided by n (corresponding to excel function: stdevp);
In the case of sampling (that is, estimating the variance of the sample), divide by (n- 1) within the root sign (corresponding to excel function: stdev);
Because we are exposed to a large number of samples, we usually use root division (n- 1).
Formula meaning
Normal distribution diagram
Subtract the sum of squares of all numbers, divide the result by the number of this group (or the number MINUS one, that is, the variance), and then find the root. The number is the standard deviation of this group of data.
The dark blue region is the range of values within one standard deviation of the average value. In normal distribution, this range accounts for 68% of all values. For normal distribution, the ratio within two standard deviations (dark blue and blue) adds up to 95%. For normal distribution, the proportion within plus or minus three standard deviations (dark blue, blue and light blue) is 99%.