# Standard Deviation Applied to Real Life Data

The concpet of standard deviation is applied to real life data through examples and their solutions.

## Example 1 - Comparing the Standard Deviation of the Percentage change of the Dow, Nasdaq and S&P during January 2022

The three data sets in the table below represents the Dow, Nasdaq and S&P indices during the month of January 2022.

a) Find the daily percentage change for each index.
b) Graph the the percent changes of the three indices in the system of axes.
c) From the graphs, which of the three indices changed most during the month of January 2022?
c) Calculate the standard deviation of the percent change of each index and verify your answer to part c).

Solution to Example 1
a)
Excel is used for the calculation of percentage change for each index.
The daily percentage change from day $d - 1$ to day $d$ is calculated using the formula
$100 \times \dfrac{ \text{Index Value of day} \; \; d - \text{Index Value of day} \;\;(d - 1)}{ \text{Index Value of day} \;\; (d - 1)}$
In the examples below, the values of the indices are in columns B,C and D and the calculated percentage change will be in columns E, F a,d G.
The Dow starts in cell B2, type the formula "=100*(B3-B2)/B2" in E3, and then copy the formula in the cell E3 in the cells E4, E5, ....
The Nasdaq starts in C2, type the formula "=100*(C3-C2)/C2" in cell F3, and then copy the formula in the cell F3 in the cells F4, F5, ....
The S&imp;P starts in D2, type the formula "=100*(D3-D2)/D2" in G3, and then copy the formula in the cell G3 in the cells G4, G5, ....
The results are shown below.

b)
The graphs of the percentage changes of all three indices is shown below.

c)
From the graphs above, the Nasdaq has changed most during the month of January 2022.
d)
One way to measure the changes in a given data set is to calculate and compare the standard deviation of the percentage change of all three indices.
Below is shown the way to calculate the standard deviation using Excel.
Use the function "=STDEV.P(E3:E20)" in a cell below the data of the percentage change of Dow index which is in column E and then press "Enter".
Use the function "=STDEV.P(F3:F20)" in a cell below the data of the percentage change of Nasdaq index which is in column F and then press "Enter".
Use the function "=STDEV.P(G3:G20)" in a cell below the data of the percentage change of S&imp;P index which is in column G and then press "Enter".

The results of the standard deviation of all three indices is shown below and we can confirm that the Nasdaq standard deviation is the largest as explained in part c)

## Example 2 - Comparing the Standard Deviation of the Nasdaq and its Moving Average

The data below is part of data related to the Nasdaq index and its moving average 02/04/2021 to 02/02/2022. The whole data set may downloaded at Nasdaq Index and Moving Average February 2021 to February 2022

a) Use the steps in Descriptive Statistics Using Excel to calculate mean median , standard deviation and range of the Nasdaq and its moving average from the 03/05/2021 (because there is no data for the moving average prior to this date) and compare them.

Solution to Example 2
The results are shown in the top right of the table below.

We now group the statistics that we need to compare on a table and calculate the difference in percentage as follows:
Nasdaq Moving Average Difference in Percent
Mean 14555.87644 14526.41737 0.2%
Median 14654.01953 14692.2486 - 0.2%
Standard Deviation 818.0556152 786.078655 4.0%
Range 3448.280274 2675.168527 25.25%
The moving average tend to smooth the data which explains why the two statistics that measure variations such as the standard deviation and range get smaller as we average the data over a given period.

## More References and links

1. Complete Business Statistics - Amir D. ACZEL and JAYAVEL SOUNDERPANDIAN - 6th International Edition - 2006 - ISBN 007 - 124416-6
2. Solutions for Elementary Statistics a Step by Step Approach - Allan G. Bluman - 9th Edition - 2017 - ISBN-10 : 1259755339
3. Complete Business Statistics - Amir D. ACZEL - 2009 - ISBN-10 : 0073373605
4. Statistics - James McClave et Terry Sincich - 13th Edition - 2016 - ISBN-10 : 0134080211
5. Variance and Standard Deviation
6. Measures of Central Tendency