Basic Statistics

Introduction to statistics

MCQs Skewness and Kurtosis 11

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This quiz contains MCQs Skewness and Kurtosis covering the shape of the distribution, Measures of central tendency such as mean, median, mode, Weighted mean, data and type of data, sources of data, Measures of Dispersion/ Variation, Standard Deviation, Variance, Range, and measure of position, etc. Let us start the MCQs Skewness and Kurtosis Quiz.

1. If the frequency curve has a longer tail to the left, the distribution is

 
 
 
 

2. If the moment Ratio $\beta_2=3$ then the distribution is

 
 
 
 

3. The distribution is positively skewed if

 
 
 
 

4. The degree to which numerical data tend to spread out about an average value is called

 
 
 
 

5. In symmetrical distribution if $Q_1=4, Q_3=12$ then median is

 
 
 
 

6. For a symmetrical distribution

 
 
 
 

7. When a distribution is symmetrical and has one mode, the highest point on the curve is called the

 
 
 
 

8. In a symmetrical distribution, the mean is 25. What is the value of the median?

 
 
 
 

9. In Symmetrical distribution $Q_3-Q_1=20$, Median = 15, $Q_3$ is equal to

 
 
 
 

10. When mean, median, and mode are identical the distribution is

 
 
 
 

11. The first three moments of a distribution about the mean $\overline{X}$ are 1, 4, and 0. The distribution is

 
 
 
 

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Basic statistics deals with the measure of central tendencies (such as mean, median, mode, weighted mean, geometric mean, and Harmonic mean) and measures of dispersion (such as range, standard deviation, and variances).

Basic statistical methods include planning and designing the study, collecting data, arranging, and numerical and graphically summarizing the collected data.

Basic statistics are also used to perform statistical analysis to draw meaningful inferences.

MCQs Skewness and Kurtosis

A basic visual inspection of data using some graphical and numerical statistics may reveal useful hidden information already available in the data. The graphical representation includes a bar chart, pie chart, dot chart, box plot, etc.

Companies related to finance, communication, manufacturing, charity organizations, government institutes, simple to large businesses, etc. are all examples that have a massive interest in collecting data and measuring different sorts of statistical findings. This helps them to learn from the past, noticing the trends, and planning for the future.

Various graphical and numerical ways are used to check the skewness of the data. Skewness is the lack of symmetry in distribution.

MCQs Skewness and Kurtosis

  • When a distribution is symmetrical and has one mode, the highest point on the curve is called the
  • If the moment Ratio $\beta_2=3$ then the distribution is
  • For a symmetrical distribution
  • The degree to which numerical data tend to spread out about an average value is called
  • In symmetrical distribution if $Q_1=4, Q_3=12$ then median is
  • The first three moments of a distribution about the mean $\overline{X}$ are 1, 4, and 0. The distribution is
  • The distribution is positively skewed if
  • In Symmetrical distribution $Q_3-Q_1=20$, Median = 15, $Q_3$ is equal to
  • When mean, median, and mode are identical the distribution is
  • In a symmetrical distribution, the mean is 25. What is the value of the median?
  • If the frequency curve has a longer tail to the left, the distribution is

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Introduction to R Language

Measures of Dispersion: Variance (2021)

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Variance is one of the most important measures of dispersion of a distribution of a random variable. The term variance was introduced by R. A. Fisher in 1918. The variance of a set of observations (data set) is defined as the mean of the squares of deviations of all the observations from their mean. When it is computed for the entire population, the variance is called the population variance, usually denoted by $\sigma^2$, while for sample data, it is called sample variance and denoted by $S^2$ to distinguish between population variance and sample variance. Variance is also denoted by $Var(X)$ when we speak about the variance of a random variable. The symbolic definition of population and sample variance is

$\sigma^2=\frac{\sum (X_i – \mu)^2}{N}; \quad \text{for population data}$

$\sigma^2=\frac{\sum (X_i – \overline{X})^2}{n-1}; \quad \text{for sample data}$

It should be noted that the variance is in the square of units in which the observations are expressed and the variance is a large number compared to the observations themselves. The variance because of its nice mathematical properties, assumes an extremely important role in statistical theory.

Variance can be computed if we have standard deviation as the variance is the square of standard deviation i.e. Variance = (Standard Deviation)$^2$.

measures-of-dispersion

Variance can be used to compare dispersion in two or more sets of observations. Variance can never be negative since every term in the variance is the squared quantity, either positive or zero.
To calculate the standard deviation one has to follow these steps:

  1. First, find the mean of the data.
  2. Take the difference of each observation from the mean of the given data set. The sum of these differences should be zero or near zero it may be due to the rounding of numbers.
  3. Square the values obtained in step 1, which should be greater than or equal to zero, i.e. should be a positive quantity.
  4. Sum all the squared quantities obtained in step 2. We call it the sum of squares of differences.
  5. Divide this sum of squares of differences by the total number of observations if we have to calculate population standard deviation ($\sigma$). For sample standard deviation (S) divide the sum of squares of differences by the total number of observations minus one i.e. degree of freedom.
    Find the square root of the quantity obtained in step 4. The resultant quantity will be the standard deviation for the given data set.
Measures of Dispersion

The major characteristics of the variances are:
a)    All of the observations are used in the calculations
b)    Variance is not unduly influenced by extreme observations
c)    The variance is not in the same units as the observation, the variance is in the square of units in which the observations are expressed.

Consider a scenario: Imagine two groups of students both score an average of 70% on an exam. However, in Group A, most scores are clustered around 70%, while in Group B, scores are spread out widely. The measure of spread (like standard deviation or variance) helps distinguish these scenarios, providing a more nuanced understanding of student performance.

By understanding how spread out (scatterness of) the data points are from the average value (mean), standard deviation offers valuable insights in various practical scenarios. It allows for data-driven decision making in quality control, investment analysis, scientific research, and other fields.

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R Programming Language

Levels of Measurement (2021)

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Levels of Measurement (Scale of Measure)

The levels of measurement (scale of measures) have been classified into four categories. It is important to understand these measurement levels since they play an important part in determining the arithmetic and different possible statistical tests carried on the data. The scale of measure is a classification that describes the nature of the information within the number assigned to a variable. In simple words, the level of measurement determines how data should be summarized and presented.

It also indicates the type of statistical analysis that can be performed. The four-level of measurements are described below:

Nominal Level of Measurement (Nominal Scale)

At the nominal level of measurement, the numbers are used to classify the data (unordered group) into mutually exclusive categories. In other words, for the nominal level of measurement, observations of a qualitative variable are measured and recorded as labels or names.

Ordinal Level of Measurement (Ordinal Scale)

In the ordinal level of measurement, the numbers are used to classify the data (ordered group) into mutually exclusive categories. However, it does not allow for a relative degree of difference between them. In other words, for the ordinal level of measurement, observations of a qualitative variable are either ranked or rated on a relative scale and recorded as labels or names.

Interval Level of Measurement (Interval Scale)

For data recorded at the interval level of measurement, the interval or the distance between values is meaningful. The interval scale is based on a scale with a known unit of measurement.

Ratio Level of Measurement (Ratio Scale)

Data recorded at the ratio level of measurement are based on a scale with a known unit of measurement and a meaningful interpretation of zero on the scale. Almost all quantitative variables are recorded on the ratio level of measurement.

Levels of Measurement

Examples of levels of measurement

Examples of Nominal Level of Measurement

  • Religion (Muslim, Hindu, Christian, Buddhist)
  • Race (Hispanic, African, Asian)
  • Language (Urdu, English, French, Punjabi, Arabic)
  • Gender (Male, Female)
  • Marital Status (Married, Single, Divorced)
  • Number plates on Cars/ Models of Cars (Toyota, Mehran)
  • Parts of Speech (Noun, Verb, Article, Pronoun)

Examples of Ordinal Level of Measurement

  • Rankings (1st, 2nd, 3rd)
  • Marks Grades (A, B, C, D)
  • Evaluations such as High, Medium, and Low
  • Educational level (Elementary School, High School, College, University)
  • Movie Ratings (1 star, 2 stars, 3 stars, 4 stars, 5 stars)
  • Pain Ratings (more, less, no)
  • Cancer Stages (Stage 1, Stage 2, Stage 3)
  • Hypertension Categories (Mild, Moderate, Severe)

Examples of Interval Levels of Measurement

  • Temperature with Celsius scale/ Fahrenheit scale
  • Level of happiness rated from 1 to 10
  • Education (in years)
  • Standardized tests of psychological, sociological, and educational discipline use interval scales.
  • SAT scores

Examples of Ratio Level of Measurement

  • Height
  • Weight
  • Age
  • Length
  • Volume
  • Number of home computers
  • Salary

In essence, levels of measurement act like a roadmap for statistical analysis. They guide us in selecting the most appropriate methods to extract valuable insights from the data under study. The level of measures is very important because they help us in

  • Choosing the right statistical tools: Different levels of measurement are used for different statistical methods. For example, One can compute a measure of central tendency (such as mean and median) for data on income (which is interval level), but a measure of central tendency (such as mean and median) cannot be computed for data on favorite color (which is nominal level, the mode can be computed regarding the measure of central tendency).
  • Drawing valid conclusions: If the statistical test is misused because of a misunderstanding of the measurement level of the data, the conclusions might be misleading or even nonsensical. Therefore, measurement levels help us ensure that analysis reflects the actual characteristics of the data.
  • Making meaningful comparisons: Levels of measurement also allow us to compare data points appropriately. For instance, one can say someone is 2 years older than another person (ordinal data), but one cannot say that their preference for chocolate ice cream is twice as strong (nominal data).
Levels of Measurement

FAQS About Levels of Measurements

  1. What do you mean by measurement levels?
  2. What is the role of Levels of Measurement in Statistics?
  3. Compare, nominal, ordinal, ratio, and interval scale.
  4. What measures of central tendency can be performed on which measurement level?
  5. What is the importance of measurement levels?
  6. Give at least five, five examples of each measurement level.

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