What is standard deviation?
The standard deviation is a single statistic measure which estimates the dispersion of the dataset. It is the measure of dispersion that gets used with the mean in order to calculate the amount of variability characterizing a set of data.
Lower numbers mean the data has a department about the mean, the values in the dataset, the degree of dispersion is relatively small. However, increased figures indicate that the values are more dispersed away from the mean. The values of data distribution turn out to be more spread out and the probability of deriving an extreme item rises.
Scatter plot that illustrates two distributions of the data having greater and lesser extents of variability. The standard deviation is expressed in the units of the original data making it more comprehensible. That is why this measure of variability is used most frequently. In case, a pizza restaurant records the time it takes to deliver pizzas in minutes and has an SD of 5.
In that case, the interpretation is that the delivery seems to happen usually within five-to-five minutes of the mean time. Statisticians often report the standard deviation with the mean: 20 minutes (StDev 5).
But if another pizza restaurant consists a standard deviation of delivery time 10 minutes, we can conclude that the delivery service is even less consistent. We will examine this example in more detail later in the chapter!).
So, in this post you’ll find out why you can’t live without a standard deviation, how to interpret it on a practical example, and how to calculate it.
Why is the Standard Deviation Important?
Therefore, the standard deviation has to be understood. Even though the mean is used to find the center for the data distribution, it does not show how far off the data values are from this center. More SD values mean that more data points are located further from the mean. In other words, there are more extreme values needed. Extreme values are everywhere, variability is everywhere. If you go out for dinner or to a restaurant and order a preferred dish the results are not exactly the same. The time you spend driving to work changes daily. Products can look identical to a piece that was manufactured next to them on an assembly line, merely measuring slightly differently in length and width.
When variability is high, then the occurrence of other extreme values also increases which creates problems! If the restaurant meal deviates slightly from the normal meal, you will not even like the meal. If your daily drive to work departs significantly more than the time it normally should take, then you are going to be late. And, parts produced with the dimensional tolerance that is too diverse cannot function as they are supposed to.
More often than not we become alarmed at the poles than at the median. Standard deviations allow you to know about the variation and it gives very important information on the sameness of result or the lack of thereof…
Example of Using the Standard Deviation
Let’s say two pizza restaurants take similar ads claiming the delivery time to be twenty minutes. Why are we starving and at least one of us looks like we just came from a photo shoot? But we do understand that the mean does not give entire information of the data set! To make our decision regarding the restaurant, we should compare the standard deviations PSD and TTSD. Suppose now we can collect their delivery time information. First restaurant has a SD of 10 minutes while the second restaurant has value of 5. The way in which this changes the Logistics Deliveries is actually explained by the next slot.
The following graphs that we are going to generate include the SDs to answer this question. The restaurant with the greater standard deviation is the one with the drivers longer delivery time (10 minutes) and their distribution curve is wider.
In these charts, we will assume that if the waiting time is more than 30 minutes, it is quite out of order – we are hungry! The blackened parts of the bars above show how often the delivery took more than 30 minutes. Still, nearly 16% of deliveries in the high variability pizza joint restaurant take more than 30 minutes while less than 2% impulses of the low variability restaurant do so. Both have a mean delivery time of 20 minutes but I sure know where I would order food from when I am hungry!
There is different procedure that you can follow on what do with the standard deviation once you’ve calculated it. The above graphs combine the SD with the normal probability distribution. However, it is possible to use the Empirical Rule or Chebyshev’s Theorem, in order to understand how the standard deviation is situated regarding the values of the distribution. You can however determine the coefficient of variation which involve the use of both the standard deviation and the mean.
Standard Deviation Formula
- s = the sample StDev
- N = number of observations
- Xi = value of each observation
- x̄ = the sample mean
Statisticians refer to the numerator portion of the standard deviation formula as the sum of squares.
Technically, this formula is for the sample standard deviation. The population version uses N in the denominator. Read my post, Measures of Variability, to learn about the differences between the population and sample varieties.
Step-by-Step Example of Calculating the Standard Deviation
Calculating the standard deviation involves the following steps. The numbers correspond to the column numbers.
The calculations take each observation (1), subtract the sample mean (2) to calculate the difference (3), and square that difference (4).
Then, at the bottom, sum the column of squared differences and divide it by 16 (17 – 1 = 16), which equals 201. Statisticians call this value the variance.
Calculate the square root of the variance to derive the SD.
FAQ’s
1. What is meant by standard deviation and why is it useful.
It assists in visualizing the dispersion of data and decision making in different areas of operation such as finance, business and education.
2. What is different of standard deviations in analyzing data from those of mean?
The mean provides maximum value or losing value of a data set while on the other hand the standard deviation gives out how spread apart from this mean the data will be.
3. What mistakes are usually made while computing standard deviation?
Mistakes occur in squaring the deviations or in forgetting the square root or in counting the data items.
4. Which one is appropriate to be used in which kind of context: standard deviation or variance?
Standard deviation is more advisable such a time as when you desire an easily understandable measurement of spread which is in the same scale as that of the values.
5. Can standard deviation be zero?
Yes, all points have to be equal, for the standard deviation to be zero.
6. In some way it (standard deviation) is used to define normal distribution.
Up in one standard deviation 68% of the data points, in two standard deviations 95% of data points and in three standard deviations 99.7% of data points lie.
7. In hypothesis testing, how can standard deviation benefit?
Standard deviation is used in hypothesis testing in order to derive test statistics such as the z-score of test results.
8. The next set of questions are: Can standard variation be negative?
No, standard deviation is always non-negative because it measures dispersion.
9. What does standard deviation do to outliers?
Standard variation is sensitive to outliers; therefore, outliers may severely raise the standard variation.
10. What is the major difference between absolute deviation and standard deviation?
Absolute deviation employs the utilization of the results from absolute differences extending from the average while on the other hand, standard deviation employs squared differences from the average.