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Questions and Answers
What is the definition of skewness?
What is the definition of skewness?
Degree of asymmetry of distribution about a mean.
Which of these terms describes a distribution that is skewed to the left?
Which of these terms describes a distribution that is skewed to the left?
A distribution can be symmetric if it can be folded along a vertical axis.
A distribution can be symmetric if it can be folded along a vertical axis.
True
What is the formula for calculating the coefficient of skewness for ungrouped data?
What is the formula for calculating the coefficient of skewness for ungrouped data?
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If Sk < 0, what term describes this distribution?
If Sk < 0, what term describes this distribution?
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What does K < 3 indicate in terms of kurtosis?
What does K < 3 indicate in terms of kurtosis?
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In which of these distributions is K = 3?
In which of these distributions is K = 3?
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The state or quality of flatness or peakedness of the curve describing a frequency distribution about its mode is called kurtosis.
The state or quality of flatness or peakedness of the curve describing a frequency distribution about its mode is called kurtosis.
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Identify the three types of skewness.
Identify the three types of skewness.
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What is the formula for calculating the coefficient of kurtosis for grouped data?
What is the formula for calculating the coefficient of kurtosis for grouped data?
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Study Notes
Course Outcome 1: Measures of Shapes
- Course name: MATH154-4 Quantitative Methods
- Department: Mathematics, Mapua University
Objectives
- Compute and interpret the coefficient of skewness and kurtosis
- Differentiate between the coefficient of skewness and kurtosis
- Identify three types of skewness
- Identify three types of kurtosis
Skewness
- Definition: The degree of asymmetry of a distribution around its mean. It measures how the data deviates from being symmetrical.
- Interpretation: Symmetric, positively skewed, or negatively skewed.
Discussion on Skewness
- Symmetric Distribution: A distribution that can be folded along a vertical axis to reveal mirroring sides.
- Skewed Distribution: A distribution that lacks symmetry with respect to a vertical axis.
Types of Skewness
- Visual representations (Figures (a), (b), and (c)) depict the three types.
- Figure (a) is skewed to the right (positive skewness) with a longer tail on the right side.
- Figure (b) is symmetrical.
- Figure (c) is skewed to the left (negative skewness) with a longer tail on the left side.
Skewness of Data
- Positive Skewness: Mode < Median < Mean
- Symmetric: Mode = Median = Mean
- Negative Skewness: Mode > Median > Mean
Formula - Ungrouped Data (Skewness)
- Sk = Σ(x - x̄)³/ns³
- Sk = coefficient of skewness
- x = score/observation
- x̄ = mean
- s = standard deviation
- n = number of observations
Formula - Grouped Data (Skewness)
- Sk = Σf(x – x̄)³/ns³
- Sk = coefficient of skewness
- x = class mark
- x̄ = mean
- s = standard deviation
- f = frequency
- n = total number of frequency
Interpretation of Skewness Values
- Sk < 0: Negatively skewed ("skewed to the left")
- Sk = 0: Symmetric
- Sk > 0: Positively skewed ("skewed to the right")
- |Sk| < 1/2: Approximately Symmetric
- 1/2 ≤ |Sk| ≤ 1: Moderately skewed
- |Sk| > 1: Highly skewed
Kurtosis
- Definition: A measure of the degree to which a unimodal distribution is peaked. It describes the state or quality of flatness or peakedness of the curve describing a frequency distribution.
Types of Kurtosis
- Leptokurtic: High peak, heavy tails
- Mesokurtic: Normal distribution (average)
- Platykurtic: Flat peak, thin tails
Formula - Ungrouped Data (Kurtosis)
- K = Σ(x - x̄)⁴/ns⁴
- K = coefficient of Kurtosis
- x = score/observation
- x̄ = mean
- s = standard deviation
- n = number of observations
Formula - Grouped Data (Kurtosis)
- K = Σf(x - x̄)⁴/ns⁴
- K = coefficient of Kurtosis
- x = class mark
- x̄ = mean
- s = standard deviation
- f = frequency
- n = total number of frequency
Interpretation of Kurtosis Values
- K < 3: Platykurtic
- K = 3: Mesokurtic
- K > 3: Leptokurtic
Example
- Data on dresses made by a factory in 10 days (52, 57, 55, 63, 50, 52, 60, 58, 54, 56)
- Calculate skewness and kurtosis, then interpret the results (Solution steps provided in the screenshots).
Summary (Skewness)
- Skewness measures asymmetry in a distribution.
- Positive skewness: mode < median < mean
- Symmetric: mode = median = mean
- Negative skewness: mode > median > mean
Summary (Kurtosis)
- Kurtosis measures the peakedness of a distribution.
- Platykurtic: K < 3
- Mesokurtic: K = 3
- Leptokurtic: K > 3
References
- (List of references from provided text)
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Description
This quiz covers the concepts of skewness and kurtosis in statistics, focusing on computation and interpretation. Learn about the different types of skewness and how they affect data distribution. Engage with visual representations to enhance your understanding of these statistical measures.
