The standard deviation is a type of "mean" and can often help you find the story behind the data. To understand this concept, it can be helpful to learn what statisticians call the normal distribution of data.

A normal distribution of data means that most of the examples in a data set are close to the "average".

The standard deviation is a statistic that indicates how closely all the different examples are clustered around the mean in a data set. If the examples are fairly close together and the bell-shaped curve is steep, the standard deviation is small. If the examples are spread out and the bell-shaped curve is relatively flat, this means that you have a relatively large standard deviation. The exact formula is described in the examples.

Calculation

Let's assume you are writing a story about nutrition. You need to look at people's typical daily calorie consumption. As with most data, the number of people would like to turn out to be normally distributed. That is, for most people, their consumption will be well below average.

The x-axis (the horizontal axis) is the value in question (for example, calories consumed or crimes committed). The y-axis (the vertical axis) is the number of data points for each value on the x-axis - in other words, the number of people eating x number of calories or the number of cities with x number of crimes committed. Now, not all data sets have graphs that look this perfect. Some have relatively flat curves, others are quite steep. Sometimes the mean leans slightly to one side or the other. However, all normally distributed data have the same shape of the "bell curve".

Example

An example of calculating the standard deviation in mathematics:

Formula:

SA = √ (∑ (r i - r avg)²) / n - 1)

  • r i : the return observed in a period
  • r avg : the arithmetic mean of the observed returns
  • n : the number of observations in the data set

An investor wants to calculate the standard deviation experience of his investment portfolio over the last four months.

Figures:

  • May / 15%
  • June / -9%
  • July / 10%
  • August / 6%

The first step is to calculate Ravg, the arithmetic mean:

(0,15 - 0,09 0,10 - 0,06) / 4 = 0,055

The arithmetic mean of the returns is 5.5%

Now we apply the formula: SA = √ (∑ (r i - r avg)²) / n - 1)

SA = √ ∑ ((0.15 - 0.055)² (0.09 0.055)² (0.10 - 0.055)² (0.06 - 0.055)²) / 3 = 0.1034

Result:

The standard deviation of the returns is 10.34%.

Explanation:

The investor now knows that the returns on his portfolio fluctuate by around 10% month-on-month. The information can be used to change the portfolio to improve the investor's attitude to risk. If the investor is risk-averse and comfortable with investing in higher-risk securities and can tolerate a higher standard deviation, they might consider adding some stocks. Conversely, a risk-averse investor may not be comfortable with this standard deviation and may want to add safer investments such as large cap stocks.

Mathematics

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