Basic Statistics and Data Analysis

Lecture notes, MCQS of Statistics

Skewness: Measure of Asymmetry

The skewed and askew are widely used terminologies that refer to something that is out of order or distorted on one side. Similarly, when referring to the shape of frequency distributions or probability distributions, the term skewness also refers to asymmetry of that distribution. A distribution with an asymmetric tail extending out to the right is referred to as “positively skewed” or “skewed to the right”, while a distribution with an asymmetric tail extending out to the left is referred to as “negatively skewed” or “skewed to the left”. The range of skewness is from minus infinity ($-\infty$) to positive infinity ($+\infty$). In simple words skewness (asymmetry) is measure of symmetry or in other words skewness is the lack of symmetry.

Karl Pearson (1857-1936) first suggested measuring skewness by standardizing the difference between the mean and the mode, such that, $skewness=\frac{\mu-mode}{\text{standard deviation}}$. Since, population modes are not well estimated from sample modes, therefore Stuart and Ord, 1994 suggested that one can estimate the difference between the mean and the mode as being three times the difference between the mean and the median. Therefore, the estimate of skewness will be: $skewness=\frac{3(M-median)}{\text{standard deviation}}$. Many of the statisticians use this measure but after eliminating the ‘3’, that is, $skewness=\frac{M-Median}{\text{standard deviation}}$. This statistic ranges from $-1$ to $+1$. According to Hilderand, 1986, absolute values of skewness above 0.2 indicate great skewness.

Skewness has also been defined with respect to the third moment about the mean, that is $\gamma_1=\frac{\sum(X-\mu)^3}{n\sigma^3}$, which is simply the expected value of the distribution of cubed $Z$ scores. Skewness measured in this way is also sometimes referred to as “Fisher’s skewness”. When the deviations from the mean are greater in one direction than in the other direction, this statistic will deviate from zero in the direction of the larger deviations. From sample data, Fisher’s skewness is most often estimated by: $g_1=\frac{n\sum z^3}{(n-1)(n-2)}$. For large sample sizes ($n > 150$), $g_1$ may be distributed approximately normally, with a standard error of approximately $\sqrt{\frac{6}{n}}$. While one could use this sampling distribution to construct confidence intervals for or tests of hypotheses about $\gamma_1$, there is rarely any value in doing so.

Arthur Lyon Bowley (1869-19570, has also proposed a measure of skewness based on the median and the two quartiles. In a symmetrical distribution, the two quartiles are equidistant from the median but in an asymmetrical distribution, this will not be the case. The Bowley’s coefficient of skewness is $skewness=\frac{q_1+q_3-2\text{median}}{Q_3-Q_1}$. Its value lies between 0 and $\pm1$.

The most commonly used measures of skewness (those discussed here) may produce some surprising results, such as a negative value when the shape of the distribution appears skewed to the right.

It is important for researchers from the behavioral and business sciences to measure skewness when it appears in their data. Great amount of skewness may motivate the researcher to investigate the existence of outliers. When making decisions about which measure of location to report and which inferential statistic to employ, one should take into consideration the estimated skewness of the population. Normal distributions have zero skewness. Of course, a distribution can be perfectly symmetric but may far away from normal distribution. Transformations of variables under study commonly employed to reduce (positive) skewness. These transformation may include square root, log, and reciprocal of variable.

For more about skewness see, Skewness

Creating Formula in Excel: Operators order of precedence

Creating Formula in Excel: Operators Order of Precedence

Creating customized (user defined) formulas in Microsoft Excel is not too difficult. For creating formulas just combine the references of your data with the correct mathematical operator (such as -, +, /, * and ^).

Microsoft Order of Precedence

The order of mathematical operations determines in which order the mathematical operations are carried out. If more than mathematical operators are used in formula, there is a specific order (sequence) that Microsoft Excel will follow to perform (compute) these mathematical operations. However, to change the order of operations, brackets (parenthesis) are used in the Excel formula. The easy way to remember the order of operations (precedence) is to remember the acronym: BEDMAS (PEDMAS), that i.e.,

The order of operations (precedence) is:

Bracket or Parenthesis
Exponents (^)
Division (/)
Multiplication (*)
Addition (+)
Subtraction (-)

Suppose, following is the screenshot of an Excel sheet. The formula is also shown in formula bar. As an example, addition (+), division (/) and multiplication (*) operators are used.

order of precedenceThe formula in screenshot performs the computation in the following order,

  • E1/F1 will be computed (answer is 1.5),
  • the answer of E1/F1 will be multiplied by value of G1 (answer is 1.5*2 = 3)
  • the answer of E1/F1 * G1 will be added to D1 (answer is 7)

Any operation(s) enclosed in brackets (parenthesis) will be carried out first followed by any exponents. After that, Excel will consider division or multiplication operations to be of equal importance. The operations are performed in the order they occur left to right in the formula. Similar sequence is also performed for addition and subtraction. Both (addition and subtraction) are considered equal in the order of operations. The operator which appears first will be computed first.

For example, see the screenshot order of precedence bracketThe sequence of operation is

  • First bracket will be computed, that is, multiplication will be performed (2 *2 = 4)
  • E1 will be divided by the answer from multiplication of F1 and G1 (3/4 = 0.75)
  • Lastly D1 will be added to the answer 0.75 (4 + 0.75 = 4.75)

Now check the sequence in the following screenshot

order of precedence bracketFor Creating formula in Excel, see the link Creating Excel Formula


Convert PDFs to Editable File Formats in 3 Easy Steps

Since the introduction of computers into our lives, we’ve been able to do things that we couldn’t do before. Slowly but surely, our PC skills have improved and today we are using new technologies that are enabling us to be better and more productive in almost every aspect of our lives.

One huge part of modern technology are digital documents that are a legacy of digital revolution. Paper documents have been replaced by digital files at one point, since they are easier to use, edit and share between colleagues and friends.

One of the most used and known digital file formats is Portable Document Format, better known as the PDF. Developed and published in the nineties, the PDF is still a number one format for managers, students, accountants, writers and many others. For more than 20 years it has been building up supporters, who use it for 3 main reasons:

  1. It’s universal — it can be opened on any device (including mobile devices).
  2. It’s shareable — documents are easily shared across all platforms.
  3. It’s standardized — the files always maintain original formatting.

Aside from attractive features that make this file format popular, there is one major downside to using PDF — the format is not so easy to edit.

If you want to make changes to your financial or project reports saved in PDF, the best thing to do is to edit your documents using a software that’s designed for that purpose. One such tool is Able2Extract Professional 11, known for its powerful and modern PDF editing features.

With Able2Extract’s integrated PDF editor you can:

  • Resize and scale more pages at once
  • Add 10 different annotations
  • Customize any individual page
  • Add and delete your PDF content
  • Extract and combine multiple PDFs
  • Redact any sensitive content

The software also converts PDF to over 10 different file formats (MS Office, AutoCAD, Image, HTML, CSV) and it’s available for all three desktop platforms.

It’s so easy to use that all you need to do is follow this three step conversion process:

  1. Click Open and select the PDF document that you want to convert.Convert PDF with Able2Extact: Open and Select PDF
  2. Select either the entire document or just a part, using the Selection panel. After making the selection, click on the desired output format.
    Convert PDF with Able2Extact: Selection Panel
  3. Choose where you want your document to be saved, and the conversion will begin.
    Convert PDF with Able2Extact: save conversion

Besides editing and conversion, the developers of Able2Extract decided to provide complete document encryption and decryption upon your PDF creation.

Now you can set up file owners, configure passwords and share your documents freely. By clicking on the “Create” button in Able2Extract, the software will automatically make a PDF document from your file.

To conclude this quick guide: the conversion of PDF files is precise, quick and most importantly — it can boost your office productivity. On the downside, the tool is aimed at experienced business professionals, with the full, lifetime license costing around $150.

To see if Able2Extract is a tool that can help you with your everyday documents struggles, you can download the free trial version. It lasts for 7 days, which is more than enough to make the right call.

See the video for further information and working of Able2Extact software


Random Walk Model

The random walk model is widely used in the area of finance. The stock prices or exchange rates (Asset prices) follow a random walk. A common and serious departure from random behavior is called a random walk (non-stationary), since today’s stock price is equal to yesterday stock price plus a random shock.

There are two types of random walks

  1. Random walk without drift (no constant or intercept)
  2. Random walk with drift (with a constant term)


A time series said to follow a random walk if the first differences (difference from one observation to the next observation) are random.

Note that in a random walk model, the time series itself is not random, however, the first differences of time series are random (the differences changes from one period to the next).

A random walk model for a time series $X_t$ can be written as

\[X_t=X_{t-1}+e_t\, \, ,\]

where $X_t$ is the value in time period $t$, $X_{t-1}$ is the value in time period $t-1$ plus a random shock $e_t$ (value of error term in time period $t$).

Since the random walk is defined in terms of first differences, therefore, it is easier to see the model as

\[X_t-X_{t-1}=e_t\, \, ,\]

where the original time series is changed to a first difference time series, that is the time series is transformed.

The transformed time series:

  • Forecast the future trends to aid in decision making
  • If time series follows random walk, the original series offers little or no insights
  • May need to analyze first differenced time series

Consider a real-world example of daily US-dollar-to-Euro exchange rate. A plot of entire history (of daily US-dollar-to-Euro exchange rate) from January 1, 1999, to December 5, 2014 looks like

Random Walk modelThe historical pattern from above plot looks quite interesting, with many peaks and valleys. The plot of the daily changes (first difference) would look like

Random Walk Model first differenceThe volatility (variance) has not been constant over time, but the day-to-day changes are almost completely random.

Remember that, random walk patterns are also widely found elsewhere in nature, for example, in the phenomenon of Brownian Motion that was first explained by Einstein.

Changing the data and creating Formula in MS-Excel

Changing the data

Before writing your required formula, you need numeric data in different columns or rows of Excels’ sheet. Suppose you want to enter few numbers in a column. Before entry these number you should first confirm the cell reference where you need to enter the data. Let start by entry number in Excels’ cell A1 and A2. For this purpose follow steps given below

  1. Click on the cell A1
  2. Type 3 from keyboard
  3. Press the ENTER or DOWN ARROW key on the keyboard. You will be in Cell A2
  4. Now type say 2 from keyboard and press ENTER key

Suppose you want to add these number in Cell C1. You need to write a formula in cell C1. After writing correct formula the content of Cell C1 will immediately changes to addition of two numbers typed in A1 and A2 and used in C1 as formula content.


Creating Formula in MS-Excel

In Excel, each formula begins with a equal sign (=), see the picture below


Therefore, when creating formulas in Excel, ALWAYS start by typing the equal sign. Equal sign is typed in the Cell where you want the answer to appear. Like image above, follow these steps

  1. Click on cell C1 with ARROW keys from keyboard or with mouse pointer.
  2. Type the equal sign in cell C1.

After typing the equal sign in step 2, you have two choices for adding cell references to the spreadsheet formula. Note that cell reference is the name of cell you want to use in formula. A1 and Aexcel-data-and-formula2 are cell references of numbers 3 and 2, respectively.

  1. You can type these references in or,
  2. You can use an Excel feature called Pointing

Pointing allows you to click with your mouse on the cell contain the data or approaching to a cell reference using keyboard ARROW keys containing your data to add. This will add cell reference toexcel-data-and-formula the formula.

After typing an equal sign in cell C3 in step 2:

  1. Click on cell A1 with the mouse pointer to enter the cell reference into the formula
  2. Type a plus (+) sign. You can also use other operators such as for multiplication use you have to use * symbol, for division / symbol and for subtraction use – etc.
  3. Click on cell A2 with the mouse pointer to enter the cell reference into the formula
  4. Press the ENTER key on the keyboard

The answer 5 should appear in cell C1.

Note if you have more than one row or column of data then you need to perform calculations on each row or column cell. It is often possible to copy the first formula to other cells. The easiest way to do this is to copy formulas with the file handle.


See also Creating Formula in Microsoft Excel


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