Basic Statistics

Introduction to statistics

A Short History of Statistics (2020)

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Here we will discuss the short History of Statistics. The word statistics was first used by a German scholar Gottfried Achenwall in the middle of the 18th century as the science of statecraft concerning the collection and use of data by the state.

The word statistics comes from the Latin word “Status” or Italian word “Statistia” or German word “Statistik” or the French word “Statistique”; meaning a political state, and originally meant information useful to the state, such as information about sizes of the population (human, animal, products, etc.) and armed forces.

According to pioneer statistician Yule, the word statistics occurred at the earliest in the book “The Element of universal erudition” by Baron (1770). In 1787 a wider definition was used by E.A.W. Zimmermann in “A Political Survey of the Present State of Europe”. It appeared in the Encyclopedia of Britannica in 1797 and was used by Sir John Sinclair in Britain in a series of volumes published between 1791 and 1799 giving a statistical account of Scotland.

In the 19th century, the word statistics acquired a wider meaning covering numerical data of almost any subject and also interpretation of data through appropriate analysis. That’s all about the short history of Statistics. Now let us see how statistics is being used in different meanings nowadays.

Brief History of Statistics

Brief History of Statistics

Now statistics is being used with different meanings.

  • Statistics refers to “numerical facts that are arranged systematically in the form of tables or charts etc. In this sense, it is always used as a plural i.e. a set of numerical information. For instance statistics on prices, road accidents, crimes, births, educational institutions, etc.
  • The word statistics is defined as a discipline that includes procedures and techniques used to collect, process, and analyze numerical data to make inferences and to reach appropriate decisions in situations of uncertainty (uncertainty refers to incompleteness, it does not imply ignorance). In this sense word statistic is used in the singular sense. It denotes the science of basing decisions on numerical data.
  • The word statistics refers to numerical quantities calculated from sample observations; a single quantity calculated from sample observations is called statistics such as the mean. Here word statistics is plural.

“We compute statistics from statistics by statistics”

History of Statistics

The first place of statistics is plural of statistics, in second place is plural sense data, and in third place is singular sense methods.

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Statistical Data: Introduction and Real Life Examples (2020)

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By statistical Data we mean, the piece of information collected for descriptive or inferential statistical analysis of the data. Data is everywhere. Therefore, everything that has past and/ or features is called statistical data.

One can find the Statistical data

  • Any financial/ economics data
  • Transactional data (from stores, or banks)
  • The survey, or census (of unemployment, houses, population, roads, etc)
  • Medical history
  • Price of product
  • Production, and yields of a crop
  • My history, your history is also statistical data

Data

Data is the plural of datum — it is a piece of information. The value of the variable (understudy) associated with one element of a population or sample is called a datum (or data in a singular sense or data point). For example, Mr. Asif entered college at the age of 18 years, his hair is black, has a height of 5 feet 7 inches, and he weighs about 140 pounds. The set of values collected for the variable from each of the elements belonging to the sample is called data (or data in a plural sense). For example, a set of 25 weights was collected from the 25 students.

Types of Data

The data can be classified into two general categories: quantitative data and qualitative data. The quantitative data can further be classified as numerical data that can be either discrete or continuous. The qualitative data can be further subdivided into nominal, ordinal, and binary data.

Qualitative data represent information that can be classified by some quality, characteristics, or criterion—for example, the color of a car, religion, blood type, and marital status.

When the characteristic being studied is non-numeric it is called a qualitative variable or an attribute. A qualitative variable is also known as a categorical variable. A categorical variable is not comparable to taking numerical measurements. Observations falling in each category (group, class) can only be counted for examples, gender (either male or female), general knowledge (poor, moderate, or good), religious affiliation, type of automobile owned, city of birth, eye color (red, green, blue, etc), etc. Qualitative variables are often summarized in charts graphs etc. Other examples are what percent of the total number of cars sold last month were Suzuki, what percent of the population has blue eyes?

Quantitative data result from a process that quantifies, such as how much or how many. These quantities are measured on a numerical scale. For example, weight, height, length, and volume.

When the variables studied can be reported numerically, the variable is called a quantitative variable. e.g. the age of the company president, the life of an automobile battery, the number of children in a family, etc. Quantitative variables are either discrete or continuous.

Statistical Data

Note that some data can be classified as either qualitative or quantitative, depending on how it is used. If a numerical is used as a label for identification, then it is qualitative; otherwise, it is quantitative. For example, if a serial number on a car is used to identify the number of cars manufactured up to that point then it is a quantitative measure. However, if this number is used only for identification purposes then it is qualitative data.

Binary Data

The binary data has only two possible values/states; such as, defected or non-defective, yes or no, and true or false, etc. If both of the values are equally important then it is binary symmetric data (for example, gender). However, if both of the values are not equally important then it can be called binary asymmetric data (for example, result: pass or fail, cancer detected: yes or no).

For quantitative data, a count will always give discrete data, for example, the number of leaves on a tree. On the other hand, a measure of a quantity will usually be continuous, for example, weigh 160 pounds, to the nearest pound. This weight could be any value in the interval say 159.5 to 160.5.

The following are some examples of Qualitative Data. Note that the outcomes of all examples of Qualitative Variables are non-numeric.

  • The type of payment (cheque, cash, or credit) used by customers in a store
  • The color of your new cell phone
  • Your eyes color
  • The make of the types on your car
  • The obtained exam grade

The following are some examples of Quantitative Data. Note that the outcomes of all examples of Quantitative Variables are numeric.

  • The age of the customer in a stock
  • The length of telephone calls recorded at a switchboard
  • The cost of your new refrigerator
  • The weight of your watch
  • The air pressure in a tire
  • the weight of a shipment of tomatoes
  • The duration of a flight from place A to B
  • The grade point average

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Top MCQs Introductory Statistics 3

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The post is about MCQs Introductory Statistics. There are 25 multiple-choice questions covering topics related to the measure of dispersions, measure of central tendency, and mean deviation. Let us start with the MCQs introductory statistics quiz with answers.

Online MCQs about Basic Statistics with Answers

1. If $X$ and $Y$ are independent then $SD(X-Y)$ is

 
 
 
 

2. Mean Deviation, Variance, and Standard Deviation of the values 4, 4, 4, 4, 4, 4 is

 
 
 
 
 

3. The measure of dispersion is changed by a change of

 
 
 
 

4. The mean deviation of the values, 18, 12, and 15 is

 
 
 
 

5. The percentage of values lies between $\overline{X}\pm 2 SD\,$ is

 
 
 
 
 

6. If $Y=-8X-5$ and SD of $X$ is 3, then SD of $Y$ is

 
 
 
 
 

7. $Var(2X+3)\,$ is

 
 
 
 

8. The standard deviation is always _________ than the mean deviation

 
 
 
 

9. The sum of squared deviations of a set of $n$ values from their mean is

 
 
 
 

10. Standard deviation is calculated from the Harmonic Mean (HM)

 
 
 
 

11. Suppose for 40 observations, the variance is 50. If all the observations are increased by 20, the variance of these increased observations will be

 
 
 
 

12. The measure of Dispersion can never be

 
 
 
 

13. The variance of a constant is

 
 
 
 

14. Variance remains unchanged by the change of

 
 
 
 

15. The sum of squares of deviation is least if measured from

 
 
 
 

16. The variance of 5 numbers is 10. If each number is divided by 2, then the variance of new numbers is

 
 
 
 
 

17. If all values are the same then the measure of dispersion will be

 
 
 
 
 

18. The lowest value of variance can be

 
 
 
 
 

19. A measure of dispersion is always

 
 
 
 

20. The range of the values -5, -8, -10, 0, 6, 10 is

 
 
 
 

21. If the standard deviation of the values 2, 4, 6, and 8 is 2.58, then the standard deviation of the values 4, 6, 8, and 10 is

 
 
 
 
 

22. Which of these is a relative measure of dispersion

 
 
 
 

23. If $a$ and $b$ are two constants, then $Var(a + bX)\,$ is

 
 
 
 
 

24. For the symmetrical distribution, approximately 68% of the cases are included between

 
 
 
 

25. Variance is always calculated from

 
 
 
 
 

MCQs Introductory Statistics with Answers

MCQs Introductory Statistics with Answers
  • A measure of dispersion is always
  • Which of these is a relative measure of dispersion
  • The measure of spread/dispersion is changed by a change of
  • Mean Deviation, Variance, and Standard Deviation of the values 4, 4, 4, 4, 4, 4 is
  • The mean deviation of the values, 18, 12, and 15 is
  • The sum of squares of deviation is least if measured from
  • The sum of squared deviations of a set of $n$ values from their mean is
  • Variance is always calculated from
  • The lowest value of variance can be
  • The variance of a constant is
  • Variance remains unchanged by the change of
  • $Var(2X+3)\,$ is
  • If $a$ and $b$ are two constants, then $Var(a + bX)\,$ is
  • Suppose for 40 observations, the variance is 50. If all the observations are increased by 20, the variance of these increased observations will be
  • Standard deviation is calculated from the Harmonic Mean (HM)
  • The variance of 5 numbers is 10. If each number is divided by 2, then the variance of new numbers is
  • If $X$ and $Y$ are independent then $SD(X-Y)$ is
  • If $Y=-8X-5$ and SD of $X$ is 3, then SD of $Y$ is
  • The standard deviation is always ———– than the mean deviation
  • If the standard deviation of the values 2, 4, 6, and 8 is 2.58, then the standard deviation of the values 4, 6, 8, and 10 is
  • For the symmetrical distribution, approximately 68% of the cases are included between
  • The percentage of values lies between $\overline{X}\pm 2 SD\,$ is
  • The measure of Dispersion can never be
  • If all values are the same then the measure of dispersion will be
  • The range of the values -5, -8, -10, 0, 6, 10 is
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Standard Deviation: A Measure of Dispersion (2017)

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The standard deviation is a widely used concept in statistics and it tells how much variation (measure of spread or dispersion) is in the data set. It can be defined as the positive square root of the mean (average) of the squared deviations of the values from their mean.
To calculate the standard deviation one has to follow these steps:

Calculation of Standard Deviation

  1. First, find the mean of the data.
  2. Take the difference of each data point from the mean of the given data set (which is computed in step 1). Note that, the sum of these differences must be equal to zero or near to zero due to rounding of numbers.
  3. Now compute the square of the differences obtained in Step 2, it would be greater than zero, and it will be a positive quantity.
  4. Now add up all the squared quantities obtained in step 3. We call it the sum of squares of differences.
  5. Divide this sum of squares of differences (obtained in step 4) by the total number of observations (available in data) if we have to calculate population standard deviation ($\sigma$). If you want t to compute sample standard deviation ($S$) then divide the sum of squares of differences (obtained in step 4) by the total number of observations minus one ($n-1$) i.e. the degree of freedom. Note that $n$ is the number of observations available in the data set.
  6. Find the square root (also known as under root) of the quantity obtained in step 5. The resultant quantity in this way is known as the standard deviation (SD) for the given data set.

The sample SD of a set of $n$ observation, $X_1, X_2, \cdots, X_n$ denoted by $S$ is

\begin{aligned}
\sigma &=\sqrt{\frac{\sum_{i=1}^n (X_i-\overline{X})^2}{n}}; Population\, SD\\
S&=\sqrt{ \frac{\sum_{i=1}^n (X_i-\overline{X})^2}{n-1}}; Sample\, SD
\end{aligned}

The standard deviation can be computed from variance too.

The real meaning of the standard deviation is that for a given data set 68% of the data values will lie within the range $\overline{X} \pm \sigma$ i.e. within one standard deviation from the mean or simply within one $\sigma$. Similarly, 95% of the data values will lie within the range $\overline{X} \pm 2 \sigma$ and 99% within $\overline{X} \pm 3 \sigma$.

Standard Deviation

Examples

A large value of SD indicates more spread in the data set which can be interpreted as the inconsistent behaviour of the data collected. It means that the data points tend to be away from the mean value. For the case of smaller standard deviation, data points tend to be close (very close) to the mean indicating the consistent behavior of the data set.

The standard deviation and variance are used to measure the risk of a particular investment in finance. The mean of 15% and standard deviation of 2% indicates that it is expected to earn a 15% return on investment and we have a 68% chance that the return will be between 13% and 17%. Similarly, there is a 95% chance that the return on the investment will yield an 11% to 19% return.

measures-of-dispersion

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