Basic Statistics

Introduction to statistics

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 
 

4. The lowest value of variance can be

 
 
 
 
 

5. Which of these is a relative measure of dispersion

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

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

 
 
 
 

10. Variance is always calculated from

 
 
 
 
 

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

 
 
 
 
 

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

 
 
 
 

14. Variance remains unchanged by the change of

 
 
 
 

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

 
 
 
 
 

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

 
 
 
 

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

 
 
 
 
 

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

 
 
 
 

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

 
 
 
 
 

20. The measure of Dispersion can never be

 
 
 
 

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. The variance of a constant is

 
 
 
 

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

 
 
 
 

24. A measure of dispersion is always

 
 
 
 

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

 
 
 
 
 

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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Skewness in Statistics A Measure of Asymmetry (2017)

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The article is about Skewness in Statistics, a measure of asymmetry. Skewed and skew 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 the 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”. In contrast, a distribution with an asymmetric tail extending out to the left is “negatively skewed” or “skewed to the left”.

Skewness in Statistics A measure of Asymmetry

Skewness in Statistics

It ranges from minus infinity ($-\infty$) to positive infinity ($+\infty$). In simple words, skewness (asymmetry) is a measure of symmetry, or in other words, skewness is a lack of symmetry.

Skewness by Karl Pearson

Karl Pearson (1857-1936) first suggested measuring skewness by standardizing the difference between the mean and the mode, such that, $\frac{\mu-mode}{\text{standard deviation}}$. 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 $$\frac{3(M-median)}{\text{standard deviation}}$$. Many of the statisticians use this measure but after eliminating the ‘3’, that is, $$\frac{M-Median}{\text{standard deviation}}$$. This statistic ranges from $-1$ to $+1$. According to Hildebrand, 1986, absolute values above 0.2 indicate great skewness.

Fisher’s Skewness

Skewness has also been defined concerning 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, 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.

Bowleys’ Coefficient of Skewness

Arthur Lyon Bowley (1869-19570, has also proposed a measure of asymmetry 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 $$\frac{q_1+q_3-2\text{median}}{Q_3-Q_1}$$. Its value lies between 0 and $\pm1$.

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

Impact of Lack of Symmetry

Researchers from the behavioral and business sciences need to measure the lack of symmetry when it appears in their data. A great amount of asymmetry may motivate the researcher to investigate the existence of outliers. When making decisions about which measure of the 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 be far away from the normal distribution. Transformations of variables under study are commonly employed to reduce (positive) asymmetry. These transformations may include square root, log, and reciprocal of a variable.

In summary, by understanding and recognizing how skewness affects the data, one can choose appropriate analysis methods, gain more insights from the data, and make better decisions based on the findings.

FAQs About Skewness

  1. What statistical measure is used to find the asymmetry in the data?
  2. Define the term Skewness.
  3. What is the difference between symmetry and asymmetry concept?
  4. Describe negative and positive skewness.
  5. What is the difference between left-skewed and right-skewed data?
  6. What is a lack of symmetry?
  7. Discuss the measure proposed by Karl Pearson.
  8. Discuss the measure proposed by Bowley’s Coefficient of Skewness.
  9. For what distribution, the skewness is zero?
  10. What is the impact of transforming a variable?

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