What is standard deviation in excel. How to calculate the percentage of deviation in Excel using two formulas

Good afternoon

In this article, I decided to look at how standard deviation works in Excel using the STANDARDEVAL function. I just haven’t described or commented on statistical functions for a very long time, and also simply because it is a very useful function for those who study higher mathematics. And helping students is sacred; I know from experience how difficult it is to master. In reality, standard deviation functions can be used to determine the stability of products sold, create prices, adjust or form an assortment, and other equally useful analyzes of your sales.

Excel uses several variations of this variance function:

Mathematical theory

First, a little about the theory, how you can describe the standard deviation function in mathematical language for using it in Excel, for analyzing, for example, sales statistics data, but more on that later. I warn you right away, I will write a lot of incomprehensible words...)))), if anything below in the text, look immediately for practical application in the program.

What exactly does standard deviation do? It estimates the standard deviation of a random variable X relative to its mathematical expectation based on an unbiased estimate of its variance. Agree, it sounds confusing, but I think students will understand what we are actually talking about!

First, we need to determine the “standard deviation”, in order to subsequently calculate the “standard deviation”, the formula will help us with this: The formula can be described as follows: standard deviation will be measured in the same units as measurements of a random variable and is used when calculating the standard arithmetic mean error, when constructing confidence intervals, when testing hypotheses for statistics, or when analyzing a linear relationship between independent variables. The function is defined as the square root of the variance of the independent variables.

Now we can define and standard deviation is an analysis of the standard deviation of a random variable X relative to its mathematical perspective based on an unbiased estimate of its variance. The formula is written like this:
I note that all two estimates are biased. In general cases, it is not possible to construct an unbiased estimate. But an estimate based on an estimate of the unbiased variance will be consistent.

Practical implementation in Excel

Well, now let’s move away from the boring theory and see in practice how the STANDARDEVAL function works. I will not consider all variations of the standard deviation function in Excel; one is enough, but in examples. As an example, let’s look at how sales stability statistics are determined.

First, look at the spelling of the function, and as you can see, it is very simple:

STANDARD DEVIATION.Г(_number1_;_number2_; ….), where:

Now let's create an example file and, based on it, consider how this function works. Since to carry out analytical calculations it is necessary to use at least three values, as in principle in any statistical analysis, I took conditionally 3 periods, this could be a year, a quarter, a month or a week. In my case - a month. For maximum reliability, I recommend taking as many periods as possible, but no less than three. All the data in the table is very simple for clarity of operation and functionality of the formula.

First, we need to calculate the average value by month. We will use the AVERAGE function for this and get the formula: = AVERAGE(C4:E4).
Now, in fact, we can find the standard deviation using the STANDARDEVAL.G function, in the value of which we need to enter the sales of the product for each period. The result will be a formula of the following form: =STANDARD DEVIATION.Г(C4;D4;E4).
Well, half the work is done. The next step is to form “Variation”, this is obtained by dividing by the average value, standard deviation and converting the result into percentages. We get the following table:
Well, the basic calculations are completed, all that remains is to figure out whether sales are stable or not. Let’s take as a condition that deviations of 10% are considered stable, from 10 to 25% are small deviations, but anything above 25% is no longer stable. To obtain the result according to the conditions, we will use the logical IF function and to obtain the result we will write the formula:

Standard deviation is one of those statistical terms in the corporate world that lends credibility to people who manage to pull it off well in a conversation or presentation, while leaving a vague sense of confusion among those who don't know what it is but are too embarrassed to ask. In fact, most managers don't understand the concept of standard deviation and if you are one of them, it's time for you to stop living a lie. In today's article, I'll tell you how this underappreciated statistical measure can help you better understand the data you're working with.

What does standard deviation measure?

Imagine that you are the owner of two stores. And to avoid losses, it is important to have clear control of stock balances. In an attempt to find out which manager manages inventory better, you decide to analyze the last six weeks of inventory. The average weekly cost of stock for both stores is approximately the same and amounts to about 32 conventional units. At first glance, the average runoff shows that both managers perform similarly.

But if you take a closer look at the activities of the second store, you will be convinced that although the average value is correct, the variability of the stock is very high (from 10 to 58 USD). Thus, we can conclude that the average does not always evaluate the data correctly. This is where standard deviation comes in.

The standard deviation shows how the values ​​are distributed around the mean in our sample. In other words, you can understand how large the spread in runoff is from week to week.

In our example, we used Excel's STDEV function to calculate the standard deviation along with the mean.

In the case of the first manager, the standard deviation was 2. This tells us that each value in the sample, on average, deviates 2 from the mean. Is it good? Let's look at the question from a different angle - a standard deviation of 0 tells us that each value in the sample is equal to its mean (in our case, 32.2). Thus, a standard deviation of 2 is not much different from 0, indicating that most values ​​are close to the mean. The closer the standard deviation is to 0, the more reliable the average. Moreover, a standard deviation close to 0 indicates little variability in the data. That is, a runoff value with a standard deviation of 2 indicates an incredible consistency of the first manager.

In the case of the second store, the standard deviation was 18.9. That is, the cost of runoff on average deviates by 18.9 from the average value from week to week. Crazy spread! The further the standard deviation is from 0, the less accurate the average is. In our case, the figure of 18.9 indicates that the average value (32.8 USD per week) simply cannot be trusted. It also tells us that weekly runoff is highly variable.

This is the concept of standard deviation in a nutshell. Although it does not provide insight into other important statistical measurements (Mode, Median...), in fact, standard deviation plays a crucial role in most statistical calculations. Understanding the principles of standard deviation will shed light on many of your business processes.

How to calculate standard deviation?

So now we know what the standard deviation number says. Let's figure out how it is calculated.

Let's look at the data set from 10 to 70 in increments of 10. As you can see, I've already calculated the standard deviation value for them using the STANDARDEV function in cell H2 (in orange).

Below are the steps Excel takes to arrive at 21.6.

Please note that all calculations are visualized for better understanding. In fact, in Excel, the calculation happens instantly, leaving all the steps behind the scenes.

First, Excel finds the sample mean. In our case, the average turned out to be 40, which in the next step is subtracted from each sample value. Each difference obtained is squared and summed up. We got a sum equal to 2800, which must be divided by the number of sample elements minus 1. Since we have 7 elements, it turns out that we need to divide 2800 by 6. From the result obtained we find the square root, this figure will be the standard deviation.

For those who are not entirely clear about the principle of calculating the standard deviation using visualization, I give a mathematical interpretation of finding this value.

Functions for calculating standard deviation in Excel

Excel has several types of standard deviation formulas. All you have to do is type =STDEV and you will see for yourself.

It is worth noting that the STDEV.V and STDEV.G functions (the first and second functions in the list) duplicate the STDEV and STDEV functions (the fifth and sixth functions in the list), respectively, which were retained for compatibility with earlier versions of Excel.

In general, the difference in the endings of the .B and .G functions indicate the principle of calculating the standard deviation of a sample or population. I already explained the difference between these two arrays in the previous article on calculating variance.

A special feature of the STANDARDEV and STANDDREV functions (the third and fourth functions in the list) is that when calculating the standard deviation of an array, logical and text values ​​are taken into account. Text and true boolean values ​​are 1, and false boolean values ​​are 0. I can't imagine a situation where I would need these two functions, so I think they can be ignored.

Method 1 Data Preparation

Method 2 Datasheet

Method 3 Calculate standard deviation

1. Place the cursor in the cell below the last value entered.
   • You can also calculate the standard deviation in any other empty cell in an Excel spreadsheet. Excel will automatically set your data range if you check the appropriate data cells.

Enter an equal sign. Remember that the formula must be entered without spaces.

2. Type "STDEV".

   This is the Excel formula for standard deviation. When you use this formula, Excel will automatically calculate the mean and standard deviation.

   • Select either STDEV (sample calculation) or STDEV (population calculation).
3. Specify the data range.
   • In Excel, data ranges are indicated as follows: (C2:C15). The entire formula will look like: "=STDEV(C2:C15)".
4. Press the "Enter" button.

   The standard deviation appears in the cell.

   • You can also use Excel's function to select the standard deviation formula. Click on "Insert Function" in the formula bar. Then select "Statistical" and select "STDEV". Enter your data range in the window that opens. Click OK.

What you will need

• Microsoft Excel
• Data
• Data table
• Formula for calculating standard deviation
• Data range

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One of the main tools of statistical analysis is the calculation of standard deviation. This indicator allows you to estimate the standard deviation for a sample or for a population. Let's learn how to use the standard deviation formula in Excel.

Determination of standard deviation

Let’s immediately determine what the standard deviation is and what its formula looks like. This quantity is the square root of the arithmetic mean of the squares of the difference between all quantities in the series and their arithmetic mean. There is an identical name for this indicator - standard deviation. Both names are completely equivalent.

But, naturally, in Excel the user does not have to calculate this, since the program does everything for him. Let's learn how to calculate standard deviation in Excel.

Calculation in Excel

You can calculate the specified value in Excel using two special functions STANDARDEVAL.V (for the sample population) and STANDARDEVAL.G (for the general population). The principle of their operation is absolutely the same, but they can be called in three ways, which we will discuss below.

Method 1: Function Wizard

1. Select a cell on the sheet where the finished result will be displayed. Click on the “Insert Function” button located to the left of the function line.
2. In the list that opens, look for the entry STANDARDDEVIATION.V or STANDARDDEVIATION.G. There is also a STANDARDEV function in the list, but it is retained from previous versions of Excel for compatibility reasons. After the entry is selected, click on the “OK” button.
3. The function arguments window opens. In each field, enter the population number. If the numbers are in sheet cells, then you can specify the coordinates of these cells or simply click on them. The addresses will be immediately reflected in the appropriate fields. After all the numbers in the population are entered, click on the “OK” button.
4. The calculation result will be displayed in the cell that was highlighted at the very beginning of the procedure for searching for the standard deviation.

Method 2: Formulas Tab

1. Select the cell to display the result and go to the “Formulas” tab.
2. In the “Function Library” tool block, click on the “Other functions” button. From the list that appears, select “Statistical”. In the next menu, we select between the values ​​of STANDARDDEVIATION.V or STANDARDDEVIATION.G, depending on whether the sample or the general population takes part in the calculations.
3. After this, the arguments window is launched. All further actions must be performed in the same way as in the first option.

Method 3: Manually entering the formula

There is also a way in which you won't need to call the arguments window at all. To do this, you must enter the formula manually.

1. Select the cell to display the result and enter an expression in it or in the formula bar according to the following template:

   STANDARD DEVIATION.Г(number1(cell_address1); number2(cell_address2);…)
   or
   =STDEV.B(number1(cell_address1); number2(cell_address2);…).

   In total, you can write up to 255 arguments if necessary.

2. After the entry is made, press the Enter button on your keyboard.

Lesson: Working with formulas in Excel

As you can see, the mechanism for calculating standard deviation in Excel is very simple. The user only needs to enter numbers from the population or references to the cells that contain them. All calculations are performed by the program itself. It is much more difficult to understand what the calculated indicator is and how the calculation results can be applied in practice. But understanding this already relates more to the field of statistics than to learning to work with software.

We are glad that we were able to help you solve the problem.

Ask your question in the comments, describing the essence of the problem in detail. Our specialists will try to answer as quickly as possible.

The mean square deviation (or standard deviation) is the second largest constant in the variation series. It is a measure of the diversity of objects included in a group and shows how much average options deviate from the arithmetic mean of the population being studied. The more scattered the options are around the average, the more often extreme or other distant classes of deviations from the average of the variation series occur, the greater the average square deviation turns out to be. Standard deviation is a measure of the variability of characteristics, due to the influence of random factors on them. Squared standard deviation ( S²) is called dispersion .

What is “random” when examined in detail? In the formula of the variant model, the random component appears in the form of a certain “additive” to the share of variants, formed under the influence of systematic factors, ± x case. . It, in turn, consists of the effects of an indefinitely large number of factors: x case . = Σ x random k.

Each of these factors may reveal its strong effect (make a large contribution), or may have almost no participation in the formation of a specific option (weak effect, insignificant contribution). Moreover, the share of the random “increase” for each option turns out to be different! Considering, for example, the size of daphnia, you can see that one individual is larger, another is smaller, because one was born several hours earlier, the other later, or one is genetically not completely identical to the others, and the third grew in a warmer zone of the aquarium, etc.

If these particular factors are not included in the controlled when collecting an option, then they, individually manifesting themselves to varying degrees, provide random variation option. The more random factors there are, the stronger they are, the further the options will be scattered around the average and the greater the variation characteristic, the standard deviation, turns out to be. In the context of our book, the term “random” is a synonym for the word “unknown”, “uncontrollable”. Until we express the intensity of a factor in some way (by grouping, gradation, number), until then it will remain a factor causing random variability.

The meaning of standard deviation (variation from the average) is expressed by the formula:

Where x- the attribute value of each object in the group,

M - arithmetic mean of the sign,

P - number of sample options.

It is more convenient to perform calculations using working formula:

,

where Σ x² - the sum of squares of the characteristic values ​​for all options,

Σ x- sum of attribute values,

n- sample volume.

For the shrew body mass example, the standard deviation would be: S= 0.897216496, and after necessary rounding S= 0.897 g.

In some cases it may be necessary to determine weighted standard deviation for a cumulative distribution composed of several samples for which the standard deviations are already known. This problem is solved using the formula:

,

Where SΣ - averaged value of standard deviation for the total distribution,

S--- averaged standard deviation values,

P - volumes of individual samples,

k- number of averaged standard deviations.

Let's consider this example. Four independent determinations of liver weight (mg) in shrews in June, July, August and September gave the following standard deviations: 93, 83, 50, 71 (at n= 17, 115, 132, 140). Substituting the required values ​​into the above formula, we obtain standard deviations for the total sample (for the entire snow-free period):

If primary statistical processing of a large number of samples is required, but not necessarily with great accuracy, you can use express method, based on knowledge of the law of normal distribution. As already noted, the extreme values ​​for the sample (with probability P= 95%) can be considered boundaries distant from the average at a distance of 2 S: x min = M − 2S, x max = M+ 2S. This means that the limit (Lim), in the range from the maximum to the minimum sample value, fits four standard deviations:

Lim = (M+ 2S) − (M − 2S) = 4S.

However, this conclusion is only valid for large samples, while corrections need to be made for small samples. The following formula for approximate calculation of standard deviation is recommended (Ashmarin et al., 1975):

,

where is the value d taken from Table 3 (against the corresponding sample size, n).

Table 3

Sample standard deviation of shrew body weight ( n= 63), calculated using the above formula, is:

S= (11.9 − 7.3) / 4 = 1.15 g,

which is quite close to the exact value, S= 0.89 g.

The use of express estimates of the standard deviation significantly reduces the time of calculations without significantly affecting their accuracy. There is only a slight tendency for the standard deviation values ​​obtained by this method to be overestimated for small sample sizes.

Standard deviation is a named value, so it can be used to compare the nature of variation of only the same characteristics. To compare the variability of heterogeneous characteristics expressed in different units of measurement, as well as to level out the influence of the measurement scale, the so-called coefficient of variation (CV), dimensionless quantity, sample estimate ratio S to own average M:

.

In our example with the body weight of a shrew:

9.6%.

Individual variability (variation) of traits is one of the most capacious characteristics of a biological population, any biological process or phenomenon. The coefficient of variation can be considered a completely adequate and objective indicator that well reflects the actual diversity of the population, regardless of the absolute value of the trait. The index was created to unify indicators of variability of different or different-sized traits by bringing them to the same scale.

Practice shows that for many biological traits there is an increase in variability (standard deviation) with an increase in their value (arithmetic mean). At the same time, the coefficient of variation remains approximately at the same level - 8-15%. As a rule, increasing differences in the distribution of a characteristic from the normal law are responsible for the increase in the coefficient of variation.

Conducting any statistical analysis is unthinkable without calculations. In this article we will look at how to calculate variance, standard deviation, coefficient of variation and other statistical indicators in Excel.

Maximum and minimum value

Average linear deviation

The average linear deviation is the average of the absolute (modulo) deviations from in the analyzed data set. The mathematical formula is:

a– average linear deviation,

X– analyzed indicator,

X̅– average value of the indicator,

n

In Excel this function is called SROTCL.

After selecting the SROTCL function, we indicate the data range over which the calculation should occur. Click "OK".

Dispersion

(module 111)

Perhaps not everyone knows what , so I’ll explain, it’s a measure that characterizes the spread of data around the mathematical expectation. However, usually only a sample is available, so the following variance formula is used:

s 2– sample variance calculated from observational data,

X– individual values,

X̅– arithmetic mean for the sample,

n– the number of values ​​in the analyzed data set.

The corresponding Excel function is DISP.G. When analyzing relatively small samples (up to about 30 observations), you should use , which is calculated using the following formula.

The difference, as you can see, is only in the denominator. Excel has a function for calculating sample unbiased variance DISP.B.

Select the desired option (general or selective), indicate the range, and click the “OK” button. The resulting value may be very large due to the preliminary squaring of the deviations. Dispersion in statistics is a very important indicator, but it is usually used not in its pure form, but for further calculations.

Standard deviation

The standard deviation (RMS) is the root of the variance. This indicator is also called standard deviation and is calculated using the formula:

by general population

by sample

You can simply take the root of the variance, but Excel has ready-made functions for standard deviation: STDEV.G And STDEV.V(for the general and sample populations, respectively).

Standard and standard deviation, I repeat, are synonyms.

Next, as usual, indicate the desired range and click on “OK”. The standard deviation has the same units of measurement as the analyzed indicator, and therefore is comparable to the original data. More on this below.

The coefficient of variation

All indicators discussed above are tied to the scale of the source data and do not allow one to obtain a figurative idea of ​​the variation of the analyzed population. To obtain a relative measure of data dispersion, use the coefficient of variation, which is calculated by dividing standard deviation on average. The formula for the coefficient of variation is simple:

There is no ready-made function for calculating the coefficient of variation in Excel, which is not a big problem. The calculation can be made by simply dividing the standard deviation by the mean. To do this, write in the formula bar:

STANDARDEV.G()/AVERAGE()

The data range is indicated in parentheses. If necessary, use the sample standard deviation (STDEV.B).

The coefficient of variation is usually expressed as a percentage, so you can frame a cell with a formula in a percentage format. The required button is located on the ribbon on the “Home” tab:

You can also change the format by selecting from the context menu after highlighting the desired cell and right-clicking.

The coefficient of variation, unlike other indicators of the scatter of values, is used as an independent and very informative indicator of data variation. In statistics, it is generally accepted that if the coefficient of variation is less than 33%, then the data set is homogeneous, if more than 33%, then it is heterogeneous. This information can be useful for preliminary characterization of the data and for identifying opportunities for further analysis. In addition, the coefficient of variation, measured as a percentage, allows you to compare the degree of scatter of different data, regardless of their scale and units of measurement. Useful property.

Oscillation coefficient

Another indicator of data dispersion today is the oscillation coefficient. This is the ratio of the range of variation (the difference between the maximum and minimum values) to the average. There is no ready-made Excel formula, so you will have to combine three functions: MAX, MIN, AVERAGE.

The coefficient of oscillation shows the extent of the variation relative to the average, which can also be used to compare different data sets.

In general, using Excel, many statistical indicators are calculated very simply. If something is not clear, you can always use the search box in the function insert. Well, Google is here to help.

Now I suggest you watch the video tutorial.

Good afternoon

In this article, I decided to look at how standard deviation works in Excel using the STANDARDEVAL function. I just haven’t described or commented on it for a very long time, and also simply because this is a very useful function for those who study higher mathematics. And helping students is sacred; I know from experience how difficult it is to master. In reality, standard deviation functions can be used to determine the stability of products sold, create prices, adjust or form an assortment, and other equally useful analyzes of your sales.

Excel uses several variations of this variance function:

Mathematical theory

First, a little about the theory, how you can describe the standard deviation function in mathematical language for using it in Excel, for analyzing, for example, sales statistics data, but more on that later. I warn you right away, I will write a lot of incomprehensible words...)))), if anything below in the text, look immediately for practical application in the program.

What exactly does standard deviation do? It estimates the standard deviation of a random variable X relative to its mathematical expectation based on an unbiased estimate of its variance. Agree, it sounds confusing, but I think students will understand what we are actually talking about!

First, we need to determine the “standard deviation”, in order to subsequently calculate the “standard deviation”, the formula will help us with this: The formula can be described as follows: it will be measured in the same units as the measurements of a random variable and is used when calculating the standard arithmetic mean error, when constructing confidence intervals, when testing hypotheses for statistics, or when analyzing a linear relationship between independent variables. The function is defined as the square root of the variance of the independent variables.

Now we can define and standard deviation is an analysis of the standard deviation of a random variable X relative to its mathematical perspective based on an unbiased estimate of its variance. The formula is written like this:
I note that all two estimates are biased. In general cases, it is not possible to construct an unbiased estimate. But an estimate based on an estimate of the unbiased variance will be consistent.

Practical implementation in Excel

Well, now let’s move away from the boring theory and see in practice how the STANDARDEVAL function works. I will not consider all variations of the standard deviation function in Excel; one is enough, but in examples. As an example, let’s look at how sales stability statistics are determined.

First, look at the spelling of the function, and as you can see, it is very simple:

STANDARD DEVIATION.Г(_number1_;_number2_; ….), where:

Now let's create an example file and, based on it, consider how this function works. Since to carry out analytical calculations it is necessary to use at least three values, as in principle in any statistical analysis, I took conditionally 3 periods, this could be a year, a quarter, a month or a week. In my case - a month. For maximum reliability, I recommend taking as many periods as possible, but no less than three. All the data in the table is very simple for clarity of operation and functionality of the formula.

First, we need to calculate the average value by month. We will use the AVERAGE function for this and get the formula: = AVERAGE(C4:E4).
Now, in fact, we can find the standard deviation using the STANDARDEVAL.G function, in the value of which we need to enter the sales of the product for each period. The result will be a formula of the following form: =STANDARD DEVIATION.Г(C4;D4;E4).
Well, half the work is done. The next step is to form “Variation”, this is obtained by dividing by the average value, standard deviation and converting the result into percentages. We get the following table:
Well, the basic calculations are completed, all that remains is to figure out whether sales are stable or not. Let’s take as a condition that deviations of 10% are considered stable, from 10 to 25% are small deviations, but anything above 25% is no longer stable. To obtain the result according to the conditions, we will use a logical one and to obtain the result we will write the formula:

IF(H4<0,1;"стабильно";ЕСЛИ(H4<0,25;"нормально";"не стабильно"))

All ranges are taken for clarity; your tasks may have completely different conditions.
To improve data visualization, when your table has thousands of positions, you should take advantage of the opportunity to apply certain conditions that you need or use to highlight certain options with a color scheme, this will be very clear.

First, select the ones for which you will apply conditional formatting. In the “Home” control panel, select “Conditional Formatting” and in the drop-down menu, select “Rules for highlighting cells” and then click the menu item “Text contains...”. A dialog box appears in which you enter your conditions.

After you have written down the conditions, for example, “stable” - green, “normal” - yellow and “unstable” - red, we get a beautiful and understandable table in which you can see what to pay attention to first.

Using VBA for the STDEV.Y function

Anyone interested can automate their calculations using macros and use the following function:

Function MyStDevP(Arr) Dim x, aCnt&, aSum#, aAver#, tmp# For Each x In Arr aSum = aSum + x "calculate the sum of the array elements aCnt = aCnt + 1 "calculate the number of elements Next x aAver = aSum / aCnt "average value For Each x In Arr tmp = tmp + (x - aAver) ^ 2 "calculate the sum of the squares of the difference between the array elements and the average value Next x MyStDevP = Sqr(tmp / aCnt) "calculate STANDARDEV.G() End Function

Function MyStDevP(Arr) Dim x , aCnt & , aSum #, aAver#, tmp# For Each x In Arr aSum = aSum + x "calculate the sum of the array elements

DEFINITION OF THE POPULATION AND

PARAMETERS BASED ON SAMPLE STATISTICS;

AVERAGE AND STANDARD DEVIATION

Determining the population mean

(general population)

The reaction time experiment described in the Appendix to Chapter 1 was based on the results of an actual experiment. They were intended to represent data that could be obtained in an experiment with full internal validity. Thus, the average reaction time to a light signal over 17 trials represented the average that could be obtained in an experiment with an unlimited number of trials.

We use the average of a limited sample sample to infer a sufficiently large (up to an unlimited) sample population. This population is called the general population. The population average of data such as BP is denoted M x. This characteristic of the population is called a parameter. The average that we actually calculated for a given sample is called statistics, and is denoted M x. Is the M x statistic the best estimate of the M x parameter that we can obtain from our sample sample? The answer is - without proof - yes. But before you decide that this is always the case, let's move on to standard deviation, where things are different.

Calculating standard deviation

Usually, in addition to the mean of the scores, we want to know something else, namely, what is the non-systematic variation of the scores from trial to trial. The most common way to measure unsystematic variation is to calculate the standard deviation.

To do this, you determine how much each estimate (i.e. X) more or less than average ( M X). Then you square each difference ( X-M X) and add them up. Following this, you divide this amount by N number of samples Finally, you take the square root of this average.

This calculation is represented by a formula using the symbol σ x to denote standard deviation:

90This formula can be shortened by introducing a small x to represent ( X-M X). Then the formula looks like this:

(2.1A)

Let's write out the data for condition A from the appendix to Chapter I and at the same time carry out the calculations indicated by the formula for σ x

Because the

ms.

91Standard Deviation Estimation

population

To determine the population mean that would be obtained in an infinite experiment, the best estimate was actually the sample mean. The situation is different with standard deviation. In any set of real samples there are fewer results with very high or very low values ​​than in the population. And since the standard deviation is a measure of the dispersion of estimates, its value determined on the basis of a sample is always less than the population parameter sigma σ x.

A more accurate estimate of the standard deviation for the population is found using the formula

(2.2)

(2.2A)

For our numerical data:

ms.

Some experiments hypothesize that behavior in one condition is more variable than in another. Then it makes more sense to compare standard deviations rather than averages. If for both conditions N the same thing, you can compare sigmas with each other. However, when N different, sigma for the condition with less N gives a more underestimated estimate of such a population parameter as standard deviation. Therefore, you should compare the two S.

The table below will help you remember these provisions and formulas.92

Task: Calculate σ x and S x for condition B.

Answer:σ B = 15.9; σ B = 16.4.