Measures of Central Tendency in Statistics

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Questions and Answers

What is the result of P(), where  represents the impossible event?

  • 0 (correct)
  • Undefined
  • 0.5
  • 1

According to the addition theorem for two events, which formula correctly represents P(A ∪ B)?

  • P(A) + P(B) - P(A ∪ B)
  • P(A) + P(B) - P(A ∩ B) (correct)
  • P(A) + P(B) + P(A ∩ B) - P(A ∪ B)
  • P(A) + P(B) + P(A ∩ B)

Which statement is true regarding discrete random variables?

  • They correspond to a set of natural numbers. (correct)
  • They can take any value within a range.
  • They can only be integers.
  • They are characterized by continuous probability density functions.

In Bayes' theorem, what does P(Ei | A) calculate?

<p>The reverse conditional probability of event Ei given A. (C)</p> Signup and view all the answers

Which formula represents the multiplication theorem for two events A and B?

<p>P(A ∩ B) = P(A) × P(B | A) (D)</p> Signup and view all the answers

What characterizes a continuous random variable?

<p>It can take any value within an interval. (C)</p> Signup and view all the answers

Total probability for any random variable must equal what value?

<p>1 (D)</p> Signup and view all the answers

Which statement about probability functions is correct?

<p>The probability density function assigns probabilities to intervals. (B)</p> Signup and view all the answers

What is the primary purpose of identifying the trend in a time-series?

<p>To understand the long-term general tendency of the series (C)</p> Signup and view all the answers

Which of the following statements about periodic variations is correct?

<p>They repeat their effects after almost a fixed interval of time. (A)</p> Signup and view all the answers

Which of the following is NOT a component of a time-series?

<p>Cross-sectional variations (D)</p> Signup and view all the answers

Seasonal variations in a time-series typically occur over what time frame?

<p>Always less than one year (D)</p> Signup and view all the answers

What characterizes irregular variations in a time-series?

<p>They are caused by unpredictable events. (A)</p> Signup and view all the answers

The term 'cyclical variations' refers to changes that occur over what kind of time intervals?

<p>Long-term, often multiple years (B)</p> Signup and view all the answers

In the context of time-series data, which method would be least useful for analyzing irregular variations?

<p>Random sampling (D)</p> Signup and view all the answers

Which example best illustrates a time-series observation?

<p>Analyzing monthly usage of electricity over the year (D)</p> Signup and view all the answers

Which of the following is NOT a requisite of a good measure of central tendency according to Prof. R.A. Fisher?

<p>It should only consider the highest values. (A)</p> Signup and view all the answers

How is the mean calculated for a frequency distribution?

<p>Mean = 1/N ∑ f_i X_i (B)</p> Signup and view all the answers

Which of the following statements about the median is correct?

<p>The median represents the middle-most value in a data set. (B)</p> Signup and view all the answers

Which measure of central tendency is least affected by extreme values?

<p>Median (D)</p> Signup and view all the answers

If a data set has an even number of observations, how is the median calculated?

<p>The median is the average of the two middle values. (A)</p> Signup and view all the answers

What is a characteristic of the mean as a measure of central tendency?

<p>It can be significantly affected by extreme values. (C)</p> Signup and view all the answers

Which property should a good measure of central tendency possess relating to sampling fluctuations?

<p>Should be least affected by fluctuations of sampling. (A)</p> Signup and view all the answers

In calculating the mean of a dataset, which of the following operations is necessary?

<p>Adding all observations together (C)</p> Signup and view all the answers

What does the standard deviation (S.D.) measure in a data set?

<p>The average of squares of deviations from the mean (D)</p> Signup and view all the answers

In calculating standard deviation for frequency distribution, which equation is used?

<p>S.D. = 1/N Σ f_i (X_i – X)^2 (B)</p> Signup and view all the answers

What is a scatter diagram primarily used for?

<p>To provide a visual representation of bivariate data (A)</p> Signup and view all the answers

Which statement best describes correlation in bivariate data?

<p>It assesses how two variables change in relation to each other. (B)</p> Signup and view all the answers

What does the variable 'r(X,Y)' represent in correlation analysis?

<p>Coefficient of correlation between X and Y (A)</p> Signup and view all the answers

Which component is NOT part of the formula for calculating the coefficient of correlation 'r(X,Y)'?

<p>N/Σ(X - X̄) (B)</p> Signup and view all the answers

When plotting a scatter diagram, which of the following axes typically represents the two variables?

<p>One variable on the horizontal axis and another on the vertical axis (D)</p> Signup and view all the answers

What information can be derived from analyzing the ratio of change in variables during correlation analysis?

<p>The strength and direction of the relationship between two variables (C)</p> Signup and view all the answers

What is the main characteristic of probability sampling methods?

<p>Each unit has a pre-assigned probability of being selected. (A)</p> Signup and view all the answers

Which of the following is an example of a non-probability sampling method?

<p>Quota Sampling (A)</p> Signup and view all the answers

Why is it often impractical to study the actual distribution of any statistic?

<p>The number of samples is usually larger than the population size. (B)</p> Signup and view all the answers

What is a key drawback of non-probability sampling methods?

<p>They can be influenced by the subjective bias of the investigator. (A)</p> Signup and view all the answers

Which sampling method gives each unit in the population an equal chance of being selected?

<p>Simple Random Sampling (C)</p> Signup and view all the answers

Which of the following sampling methods is classified as a probability sampling technique?

<p>Cluster Sampling (D)</p> Signup and view all the answers

In what way do probability sampling methods differ from non-probability sampling methods?

<p>Probability methods rely on random selection while non-probability methods do not. (D)</p> Signup and view all the answers

What type of sampling method can be affected by investigator skill level?

<p>Quota Sampling (D)</p> Signup and view all the answers

What is the purpose of interval estimation in statistics?

<p>To identify an interval where the population parameter likely lies (B)</p> Signup and view all the answers

In the context of confidence intervals, what do the terms 'a' and 'b' represent?

<p>The confidence limits of the parameter estimate (B)</p> Signup and view all the answers

Which type of statistical hypothesis specifies a single point in the parameter space?

<p>Simple parametric hypothesis (B)</p> Signup and view all the answers

What defines the confidence coefficient in interval estimation?

<p>The level of confidence expressed in probabilistic terms (A)</p> Signup and view all the answers

How are non-parametric hypotheses fundamentally different from parametric hypotheses?

<p>They are not concerned with parameters in the distribution (A)</p> Signup and view all the answers

What is the relationship between the terms 'confidence limits' and 'confidence interval'?

<p>Confidence limits are the two values that establish the confidence interval (C)</p> Signup and view all the answers

What is the primary characteristic of a composite parametric hypothesis?

<p>It includes multiple points in the parameter space (B)</p> Signup and view all the answers

Which statement accurately describes a statistical hypothesis?

<p>It serves as a statement that can be tested through statistical methods (D)</p> Signup and view all the answers

Flashcards

Measures of Central Tendency

Statistical tools used to find the central point of a dataset.

Mean

The arithmetic average of a set of numbers. Calculated by summing all values and dividing by the total count.

Mean (Frequency Distribution)

The arithmetic average of a dataset presented as a frequency distribution. Calculated by summing the product of each value and its frequency, then dividing by the total frequency.

Median

The middle value in a sorted dataset. If the dataset has an even number of values, it's the average of the two middle values.

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Requisite of a Good Measure of Central Tendency

Criteria ensuring the measure accurately reflects the center of a dataset, (e.g., easily calculable, based on all data, not affected by extremes.)

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Individual Observations (Median)

For a list of individual values, find the middle value by ordering the values, and if there are an even number of values, find the average of the two central values.

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Population Mean

The mean of all possible values of the values in a population

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Sample Mean

The mean of a subset of a population.

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Time Series

A sequence of data points collected at equal intervals over time.

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Cross-Sectional Data

Data collected from different units at the same point in time.

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Components of a Time Series

Factors that influence the variations observed in a time series.

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Trend

The long-term general tendency of a time series to increase, decrease, or remain constant.

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Periodic Variations

Repetitive patterns in a time series that occur at regular intervals.

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Seasonal Variations

Periodic variations within a year, often influenced by natural seasons or cultural events.

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Cyclical Variations

Periodic variations that occur over longer periods than a year, influenced by economic or business cycles.

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Irregular Variations

Unpredictable fluctuations in a time series that cannot be attributed to trend or periodic variations.

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Standard Deviation

A measure of how spread out a dataset is from its mean. It's calculated as the square root of the average squared deviations.

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Scatter Diagram

A visual representation of bivariate data, showing the relationship between two variables. Each point on the diagram represents a pair of observations.

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Correlation

A statistical measure of the linear association between two variables. It tells us how strongly the variables change together.

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What is a bivariate data?

Data that focuses on two variables for a specific group of objects and examines the relationship between them.

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Covariance

A measure of how two variables change together. A positive covariance means they tend to increase or decrease together, while a negative covariance means they change in opposite directions.

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Karl Pearson's Coefficient (r)

A statistical measure of linear correlation between two variables. It ranges from -1 to 1, with 1 indicating a perfect positive correlation, -1 a perfect negative correlation, and 0 no correlation.

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What does a positive correlation mean?

Two variables change in the same direction. When one increases, the other tends to increase, and vice versa.

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What does a negative correlation mean?

Two variables change in opposite directions. When one increases, the other tends to decrease, and vice versa.

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Impossible event

An event that cannot occur in a random experiment. Its probability is always 0.

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Non-happening of an event

The event that occurs when the original event does not happen. Its probability is 1 minus the probability of the original event.

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P(A ∩ B)

The probability of both events A and B occurring, also known as the joint probability of A and B.

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Addition theorem for two events

For any two events A and B, the probability of at least one of the events occurring is the sum of their individual probabilities minus the joint probability of both happening.

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Addition theorem for three events

For any three events A, B, and C, the probability of at least one of the events occurring is the sum of their individual probabilities minus the sum of joint probabilities of pairs of events, plus the joint probability of all three happening.

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Multiplication theorem on probability

The probability of both events occurring is the product of the probability of the first event and the probability of the second event given that the first event has already happened.

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Bayes' theorem

A theorem used to update the probability of an event given new information.

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Random variable

A variable whose value is a numerical outcome of a random experiment.

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Sampling Distribution

The probability distribution of a statistic, calculated from multiple samples of the same size drawn from a population.

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Probability Sampling

A method of sampling where each unit in the population has a known probability of being selected for the sample.

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Non-Probability Sampling

A method of sampling where the probability of selecting a unit is unknown or not equal for all units.

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Simple Random Sampling

A probability sampling method where each unit has an equal and independent chance of being selected.

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Stratified Random Sampling

Probability sampling where the population is divided into subgroups (strata) and random samples are drawn from each stratum.

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Systematic Sampling

Probability sampling where a unit is selected at regular intervals from the population.

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Cluster Sampling

Probability sampling where the population is divided into clusters and a random sample of clusters is selected.

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Purposive/Judgement Sampling

Non-probability sampling where units are selected based on the researcher's judgment or knowledge.

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Interval estimation

A method used to estimate the range within which an unknown population parameter likely lies.

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Confidence interval

The range of values within which we are confident the true population parameter lies, expressed as a percentage.

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Confidence limits

The two values that define the boundaries of a confidence interval.

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Confidence coefficient

The probability that the true population parameter falls within the confidence interval.

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Statistical hypothesis

A statement about the probability distribution of a random variable, which can be tested.

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Parametric hypothesis

A statistical hypothesis that makes a statement about the parameter of a random variable.

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Simple hypothesis

A parametric hypothesis specifying a single value for the parameter.

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Composite hypothesis

A parametric hypothesis specifying multiple possible values for the parameter.

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Study Notes

Measures of Central Tendency

  • Statistics deals with numerical observations, often large in size, making understanding data difficult
  • Measures of central tendency (location) are statistical tools to find the central point of data.

Requisites of a Good Measure of Central Tendency

  • Rigorously defined
  • Easily graspable and calculable
  • Based on all observations
  • Least affected by extreme values
  • Capable of further algebraic treatment
  • Least affected by sampling fluctuations

Various Measures of Central Tendency

Mean

  • Arithmetic average of observations
    • Individual observations: Mean = (X₁ + X₂ + ... + Xₙ)/n
    • Frequency distribution: Mean = (ΣfᵢXᵢ)/N, where fᵢ is the frequency of Xᵢ and N is the total frequency

Median

  • Middle-most value in a sorted set of observations
    • Odd number of observations: ((n + 1) / 2)th value
    • Even number of observations: Average of (n /2)th and ((n + 2) / 2)th values

Various Measures of Central Tendency: Mode

  • Most frequent value in a data set
    • Individual observations: Value occurring most often
    • Frequency distribution: Class interval with the highest frequency

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