**Standard Deviation And Variance**

**ACKNOWLEDGEMENTS**

This postulation is devoted to Allah, my Creator and my Master, and envoy, Mohammed (May Allah favor and give him), who showed us the motivation behind life. My country Pakistan is the hottest womb; Allama Iqbal Open University, Islamabad; my second wonderful home; My awesome guardians, who never quit giving of themselves in incalculable ways, My dearest friend, who drives me through the valley of dimness with the light of trust and support, My cherished siblings and sisters; especially my dearest sibling, who remains by me when things look disheartening, My beloved Parents: whom I can’t compel myself to quit loving. All the general population in my life who touch my heart, I commit to this research.

**ABSTRACT**

Standard deviation and variance are two basic mathematical concepts that have an important place in various parts of the financial sector, from accounting to economics to investing. Both measure the variability of figures within a data set using the mean of a certain group of numbers. They are important to help determine volatility and the distribution of returns. But there are inherent differences between the two. While standard deviation measures the square root of the variance, the variance is the average of each point from the mean. There are three main measures of Standard deviation and variance: the mode, the median, and the mean. Each of these measures describes a different indication of the typical or central value in the distribution. Standard deviation and variance is also useful when you want to compare one piece of data to the entire data set. Let’s say you received a 60% on your last psychology quiz, which is usually in the D range. You go around and talk to your classmates and find out that the average score on the quiz was 43%. In this instance, your score was significantly higher than those of your classmates. Since your teacher grades on a curve, your 60% becomes an A. Had you not known about the measures of Standard deviation and variance, you probably would have been really upset by your grade and assumed that you bombed the test. However, we can still use other measures of Standard deviation and variance even when there are outliers. Despite the existence of outliers in a distribution, the mean can still be an appropriate measure of Standard deviation and variance, especially if the rest of the data is normally distributed. If the outlier is confirmed as a valid extreme value, it should not be removed from the dataset. Several common regression techniques can help reduce the influence of outliers on the mean value.

**Introduction**

Standard deviation and variance are very useful in psychology. It lets us know what is normal or ‘average’ for a set of data. It also condenses the data set down to one representative value, which is useful when you are working with large amounts of data. Could you imagine how difficult it would be to describe the central location of a 1000-item data set if you had to consider every number individually?

Standard deviation and variance also allow you to compare one data set to another. For example, let’s say you have a sample of girls and a sample of boys, and you are interested in comparing their heights. By calculating the average height for each sample, you could easily draw comparisons between the girls and boys.

- Standard deviation and variance are two key measures commonly used in the financial sector.
- Standard deviation is the spread of a group of numbers from the mean.
- The variance measures the average degree to which each point differs from the mean.
- While standard deviation is the square root of the variance, variance is the average of all data points within a group.
- The two concepts are useful and significant for traders, who use them to measure market volatility.

Standard deviation is a statistical measurement that looks at how far a group of numbers is from the mean. Put simply, standard deviation measures how far apart numbers are in a data set.

This metric is calculated as the square root of the variance. This means you have to figure out the variation between each data point relative to the mean. Therefore, the calculation of variance uses squares because it weighs outliers more heavily than data that appears closer to the mean. This calculation also prevents differences above the mean from canceling out those below, which would result in a variance of zero.

But how do you interpret standard deviation once you figure it out? If the points are further from the mean, there is a higher deviation within the data but if they are closer to the mean, there is a lower deviation. So the more spread out the group of numbers are, the higher the standard deviation.

A variance is the average of the squared differences from the mean. To figure out the variance, calculate the difference between each point within the data set and the mean. Once you figure that out, square and average the results.

For example, if a group of numbers ranges from 1 to 10, it will have a mean of 5.5. If you square the differences between each number and the mean and find their sum, the result is 82.5. To figure out the variance:

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- Divide the sum, 82.5, by N-1, which is the sample size (in this case 10) minus 1.
- The result is a variance of 82.5/9 = 9.17.

Note that the standard deviation is the square root of the variance so that the standard deviation would be about 3.03.

Standard deviation and variance is also useful when you want to compare one piece of data to the entire data set. Let’s say you received a 60% on your last psychology quiz, which is usually in the D range. You go around and talk to your classmates and find out that the average score on the quiz was 43%. In this instance, your score was significantly higher than those of your classmates. Since your teacher grades on a curve, your 60% becomes an A. Had you not known about the measures of Standard deviation and variance, you probably would have been really upset by your grade and assumed that you bombed the test?

**Practical study**

Other than how they’re calculated, there are a few other key differences between standard deviation and variance. For one thing, the standard deviation is a statistical measure that people can use to determine how spread out numbers are in a data set. Variance, on the other hand, gives an actual value to how much the numbers in a data set vary from the mean.

Standard deviation is the square root of variance, and the variance is expressed as a percent (especially in the context of finance). As such, the standard deviation can actually be greater than the variance since the square root of a decimal will be larger (and not smaller) than the original number when the variance is less than one (1.0 or 100%). Likewise, standard deviation will be smaller than the variance when the variance is more than one (e.g., 1.2 or 120%). These two concepts are of paramount importance for both traders and investors. That’s because they are used to measure security and market volatility, which in turn plays a large role in creating a profitable trading strategy.

Standard deviation is one of the key methods that analysts, portfolio managers, and advisors use to determine risk. When the group of numbers is closer to the mean, the investment is less risky. But when the group of numbers is further from the mean, the investment is of greater risk to a potential purchaser.

Securities that are close to their means are seen as less risky, as they are more likely to continue behaving as such. Securities with large trading ranges that tend to spike or change direction are riskier.

To demonstrate how both principles work, let’s look at an example of standard deviation and variance.

Suppose you have a series of numbers and you want to figure out the standard deviation for the group. The numbers are 4, 34, 11, 12, 2, and 26. We need to determine the mean or the average of the numbers. In this case, we determine the mean by adding the numbers up and dividing it by the total count in the group:

(4 + 34 + 18 + 12 + 2 + 26) ÷ 6 = 16

So the mean is 16. Now subtract the mean from each number then square the result:

- (4 – 16)
^{2}= 144 - (34 – 16)
^{2}= 324 - (18 – 16)
^{2}= 4 - (12 – 16)
^{2}= 16 - (2 – 16)
^{2}= 196 - (26 – 16)
^{2}= 100

Now we have to figure out the average or mean of these squared values to get the variance. This is done by adding up the squared results from above, then dividing it by the total count in the group:

(144 + 324 + 4 + 16 + 196 + 100)^{ }÷ 6 = 130.67

This means we end up with a variance of 130.67. To figure out the standard deviation, we have to take the square root of the variance, which is 11.43

The simple definition of the term variance is the spread between numbers in a data set. Variance is a statistical measurement used to determine how far each number is from the mean and from every other number in the set. You can calculate the variance by taking the difference between each point and the mean. Then square and average the results.

Standard deviation measures how data is dispersed relative to its mean and is calculated as the square root of its variance. The further the data points are, the higher the deviation. Closer data points mean a lower deviation. In finance, standard deviation calculates risk so riskier assets have a higher deviation while safer bets come with a lower standard deviation.

Investors use variance to assess the risk or volatility associated with assets by comparing their performance within a portfolio to the mean. For instance, you can use the variance in your portfolio to measure the returns of your stocks. This is done by calculating the standard deviation of individual assets within your portfolio as well as the correlation of the securities you hold.

The variance of an asset may not be a reliable metric. Calculating variance can be fairly lengthy and time-consuming, especially when there are many data points involved. Variance doesn’t account for surprise events that can eat away at returns. And variance is often hard to use in a practical sense not only is it a squared value, so are the individual data points involved.

**Data collection methods **

Seventy-eight typically developing children and adults participated in this study, including twelve 6–7 year-old children (M = 7 years; 0 months, range: 6; 0–7; 10, 9 females), nineteen 8–9 year-olds (M = 9 years; 0 months, range: 8; 1–9; 10, 7 females), eighteen 10–11 year-olds (M = 11 years; 1 months, range: 10; 2–11; 11, 11 females), fifteen 12–14 year-olds (M = 12 years; 9 months, range: 12; 0–14; 1, 10 females) and fourteen adults (M = 25 years; 11 months, range: 22; 0–32; 0, 10 females). In the current study, we focused on a particular aspect of time estimation of autistic children, namely Standard deviation and variance in time interval reproduction. This tendency of quantity judgments to gravitate towards their mean value has been recently modelled within the Bayesian framework of perceptual inference. Within this framework, the magnitude of Standard deviation and variance reflects the flexible integration of noisy sensory estimates with internal representations for the mean value of stimuli to produce final judgments. In this study, we applied this Bayesian model of Standard deviation and variance in time interval reproduction to characterize performance in typical development and in autism.

Our time interval reproduction data showed that in typically developing children Standard deviation and variance effects occur at very young ages and decrease with age. These results parallel those of Sciutti, who showed a similar reduction in regression between the ages of 7 and 10 years for a spatial interval reproduction task. Our findings therefore suggest that not only are young children able to extract statistical information for the recent history of sensory input; they do so to a greater extent than older children and adults.

At a first instance, this finding seems counterintuitive. However, there is considerable evidence that abilities for statistical learning are available very early in development, even in infants and newborns. Such statistical learning abilities are thought to tap on mechanisms for experience-dependent plasticity, which are important for cognitive development, for example language acquisition. Arguably, the pattern of higher regression in younger children in our data could reflect the greater reliance upon statistical learning at younger ages.

As regression decreased with age, temporal discrimination improved, agreeing with previous results and the study of Sciutti on the spatial domain. The computational simulations combined developmental measures of Standard deviation and variance and temporal resolution to quantify the strength of prior knowledge representations and how these compared to optimal computations. The simulations suggested that typically developing children tend to perform near optimally, with the younger children using stronger (narrower) priors, consistent with their poorer temporal resolution. The integration of noisy sensory estimates with internal representations for the mean value of stimuli is therefore flexible in childhood. As Sciutti et al.7 discuss Standard deviation and variance serves to restrain the continuously decreasing sensory noise.

The data of the autistic group revealed two main facts: (1) autistic children do show regression and indeed, at higher levels than the age- and ability matched typical children; (2) but at the same time, their temporal thresholds are about twice as high as those of the matched-typical group and very similar to the 6-year-olds. This pattern implies that the use of priors in autistic children does not increase enough to compensate for the lower precision and instead remains very similar to the typical comparison group.

The elevated levels of error of reliability and error of accuracy of autistic children in the time interval reproduction task are consistent with results of other studies using time reproduction tasks. By contrast, the decreased temporal resolution of autistic children (time discrimination task) is congruent with some studies but not with other. More research is warranted in order to unify this range of inconclusive findings on temporal resolution of autistic children. The role of methodological differences – for example the range of time intervals tested, modality or whether the intervals were unfilled or filled are worth exploring in the future. The general heterogeneity of autistic population, evidenced by inconsistent findings across a range of domains should also be considered to account for variable findings.

**SWOT analysis**

Strengths and Weaknesses | The internal environment – the situation inside the company or organization | For example: factors relating to products, pricing, costs, profitability, performance, quality, people, skills, adaptability, brands, services, reputation, processes, infrastructure, etc. | Factors tend to be in the present |

Opportunities and Threats | The external environment – the situation outside the company or organization | For example: factors relating to markets, sectors, audience, fashion, seasonality, trends, competition, economics, politics, society, culture, technology, environmental, media, law, etc. | Factors tend to be in the future |

**Conclusion & Recommendations**

**Variance and Standard Deviation** are the two important measurements in statistics. Variance is a measure of how data points vary from the mean, whereas standard deviation is the measure of the distribution of statistical data. The basic difference between both is standard deviation is represented in the same units as the mean of data, while the variance is represented in squared units. Let us learn here more about both the measurements with their definitions, formulas along with an example.

**Variance**

According to layman’s words, the variance is a measure of how far a set of data are dispersed out from their mean or average value. It is denoted as ‘σ^{2}’.

### Properties of Variance

- It is always non-negative since each term in the variance sum is squared and therefore the result is either positive or zero.
- Variance always has squared units. For example, the variance of a set of weights estimated in kilograms will be given in kg squared. Since the population variance is squared, we cannot compare it directly with the mean or the data themselves.

**Standard Deviation**

The spread of statistical data is measured by the standard deviation. Distribution measures the deviation of data from its mean or average position. The degree of dispersion is computed by the method of estimating the deviation of data points. It is denoted by the symbol, ‘σ’.

### Properties of Standard Deviation

- It describes the square root of the mean of the squares of all values in a data set and is also called the root-mean-square deviation.
- The smallest value of the standard deviation is 0 since it cannot be negative.
- When the data values of a group are similar, then the standard deviation will be very low or close to zero. But when the data values vary with each other, then the standard variation is high or far from zero.

**Variance and Standard Deviation Formula**

As discussed, the variance of the data set is the average square distance between the mean value and each data value. And standard deviation defines the spread of data values around the mean.

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