text: ## Measures of Dispersion
  
   7.2 Measures of Dispersion
  
   Some formula for a grouped data $$\text{range}=\text{midpoint of the highest class} - \text{midpoint of the lowest class}$$ $$\text{Interquartile range, IQR}=Q_3-Q_1$$ \begin{aligned} \text{Mean, }\bar{x}=\frac{\sum fx}{\sum f} \end{aligned} \begin{aligned} \text{Variance, }\sigma^2=\frac{\sum fx^2}{\sum f}-\bar{x}^2 \end{aligned} \begin{aligned} \text{Standard deviation, }\sigma=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}} \end{aligned} where \begin{aligned} &x=\text{midpoint of the class interval}\\ &f=\text{frequency}\\ \end{aligned}
  
   Example 5 The frequency table below shows the volumes of water to the nearest litres, used daily by a group of families in a housing area. Calculate the variance and standard deviation of the data. Solution: \begin{aligned} \text{Mean, }\bar{x}&=\frac{\sum fx}{\sum f} \\ &=\frac{17417.5}{95}\\ &=183.34 \end{aligned} \begin{aligned} \text{Variance, }\sigma^2&=\frac{\sum fx^2}{\sum f}-\bar{x}\\ &=\frac{3 215 133.75}{95}-\bigg(\frac{17 417.5}{95}\bigg)^2\\ &=229.1856 \end{aligned} \begin{aligned} \text{Standard deviation, }\sigma&=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}}\\ &=\sqrt{229.1856} \\ &=15.1389 \end{aligned}
  
   What is a boxplot? A box plot is a method to display a group of numerical data graphically based on the five number summary of data. They are the minimum value, first quartile, median, third quartile and maximum value. Similar to the histrogram and frequency polygon, the shape of a distribution can also be identified through the box plot.
  
   Example 6 The ogive below shows the masses in g, of 90 starfruits. a) Construct a box plot based on the ogive. b) Hence, state the distribution shape of the data. Solution: a) \begin{aligned} &\text{Minimum value} = 80\\\\ &\text{Maximum value} = 150\\\\ &Q_1=116\\\\ &Q_2=123\\\\ &Q_3=128 \end{aligned} b) The distribution of the data is skewed to the left because the left side of the box plot is longer than the right side of the box plot.
  
  ## Measures of Dispersion
  
   7.2 Measures of Dispersion
  
   Some formula for a grouped data $$\text{range}=\text{midpoint of the highest class} - \text{midpoint of the lowest class}$$ $$\text{Interquartile range, IQR}=Q_3-Q_1$$ \begin{aligned} \text{Mean, }\bar{x}=\frac{\sum fx}{\sum f} \end{aligned} \begin{aligned} \text{Variance, }\sigma^2=\frac{\sum fx^2}{\sum f}-\bar{x}^2 \end{aligned} \begin{aligned} \text{Standard deviation, }\sigma=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}} \end{aligned} where \begin{aligned} &x=\text{midpoint of the class interval}\\ &f=\text{frequency}\\ \end{aligned}
  
   Example 5 The frequency table below shows the volumes of water to the nearest litres, used daily by a group of families in a housing area. Calculate the variance and standard deviation of the data. Solution: \begin{aligned} \text{Mean, }\bar{x}&=\frac{\sum fx}{\sum f} \\ &=\frac{17417.5}{95}\\ &=183.34 \end{aligned} \begin{aligned} \text{Variance, }\sigma^2&=\frac{\sum fx^2}{\sum f}-\bar{x}\\ &=\frac{3 215 133.75}{95}-\bigg(\frac{17 417.5}{95}\bigg)^2\\ &=229.1856 \end{aligned} \begin{aligned} \text{Standard deviation, }\sigma&=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}}\\ &=\sqrt{229.1856} \\ &=15.1389 \end{aligned}
  
   What is a boxplot? A box plot is a method to display a group of numerical data graphically based on the five number summary of data. They are the minimum value, first quartile, median, third quartile and maximum value. Similar to the histrogram and frequency polygon, the shape of a distribution can also be identified through the box plot.
  
   Example 6 The ogive below shows the masses in g, of 90 starfruits. a) Construct a box plot based on the ogive. b) Hence, state the distribution shape of the data. Solution: a) \begin{aligned} &\text{Minimum value} = 80\\\\ &\text{Maximum value} = 150\\\\ &Q_1=116\\\\ &Q_2=123\\\\ &Q_3=128 \end{aligned} b) The distribution of the data is skewed to the left because the left side of the box plot is longer than the right side of the box plot.
explanation: The document provides an overview of measures of dispersion, including formulas for range, interquartile range, mean, variance, and standard deviation for grouped data. It includes two examples with solutions:
  
  1. Example 5: Calculation of variance and standard deviation for the volumes of water used daily by a group of families.
  2. Example 6: Construction of a box plot and determination of the distribution shape based on an ogive of starfruit masses.
  
  The document contains questions and answers within the examples.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: ## Measures of Dispersion
  
   7.2 Measures of Dispersion
  
   Some formula for a grouped data $$\text{range}=\text{midpoint of the highest class} - \text{midpoint of the lowest class}$$ $$\text{Interquartile range, IQR}=Q_3-Q_1$$ \begin{aligned} \text{Mean, }\bar{x}=\frac{\sum fx}{\sum f} \end{aligned} \begin{aligned} \text{Variance, }\sigma^2=\frac{\sum fx^2}{\sum f}-\bar{x}^2 \end{aligned} \begin{aligned} \text{Standard deviation, }\sigma=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}} \end{aligned} where \begin{aligned} &x=\text{midpoint of the class interval}\\ &f=\text{frequency}\\ \end{aligned}
  
   Example 5 The frequency table below shows the volumes of water to the nearest litres, used daily by a group of families in a housing area. Calculate the variance and standard deviation of the data. Solution: \begin{aligned} \text{Mean, }\bar{x}&=\frac{\sum fx}{\sum f} \\ &=\frac{17417.5}{95}\\ &=183.34 \end{aligned} \begin{aligned} \text{Variance, }\sigma^2&=\frac{\sum fx^2}{\sum f}-\bar{x}\\ &=\frac{3 215 133.75}{95}-\bigg(\frac{17 417.5}{95}\bigg)^2\\ &=229.1856 \end{aligned} \begin{aligned} \text{Standard deviation, }\sigma&=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}}\\ &=\sqrt{229.1856} \\ &=15.1389 \end{aligned}
  
   What is a boxplot? A box plot is a method to display a group of numerical data graphically based on the five number summary of data. They are the minimum value, first quartile, median, third quartile and maximum value. Similar to the histrogram and frequency polygon, the shape of a distribution can also be identified through the box plot.
  
   Example 6 The ogive below shows the masses in g, of 90 starfruits. a) Construct a box plot based on the ogive. b) Hence, state the distribution shape of the data. Solution: a) \begin{aligned} &\text{Minimum value} = 80\\\\ &\text{Maximum value} = 150\\\\ &Q_1=116\\\\ &Q_2=123\\\\ &Q_3=128 \end{aligned} b) The distribution of the data is skewed to the left because the left side of the box plot is longer than the right side of the box plot.
  
  ## Measures of Dispersion
  
   7.2 Measures of Dispersion
  
   Some formula for a grouped data $$\text{range}=\text{midpoint of the highest class} - \text{midpoint of the lowest class}$$ $$\text{Interquartile range, IQR}=Q_3-Q_1$$ \begin{aligned} \text{Mean, }\bar{x}=\frac{\sum fx}{\sum f} \end{aligned} \begin{aligned} \text{Variance, }\sigma^2=\frac{\sum fx^2}{\sum f}-\bar{x}^2 \end{aligned} \begin{aligned} \text{Standard deviation, }\sigma=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}} \end{aligned} where \begin{aligned} &x=\text{midpoint of the class interval}\\ &f=\text{frequency}\\ \end{aligned}
  
   Example 5 The frequency table below shows the volumes of water to the nearest litres, used daily by a group of families in a housing area. Calculate the variance and standard deviation of the data. Solution: \begin{aligned} \text{Mean, }\bar{x}&=\frac{\sum fx}{\sum f} \\ &=\frac{17417.5}{95}\\ &=183.34 \end{aligned} \begin{aligned} \text{Variance, }\sigma^2&=\frac{\sum fx^2}{\sum f}-\bar{x}\\ &=\frac{3 215 133.75}{95}-\bigg(\frac{17 417.5}{95}\bigg)^2\\ &=229.1856 \end{aligned} \begin{aligned} \text{Standard deviation, }\sigma&=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}}\\ &=\sqrt{229.1856} \\ &=15.1389 \end{aligned}
  
   What is a boxplot? A box plot is a method to display a group of numerical data graphically based on the five number summary of data. They are the minimum value, first quartile, median, third quartile and maximum value. Similar to the histrogram and frequency polygon, the shape of a distribution can also be identified through the box plot.
  
   Example 6 The ogive below shows the masses in g, of 90 starfruits. a) Construct a box plot based on the ogive. b) Hence, state the distribution shape of the data. Solution: a) \begin{aligned} &\text{Minimum value} = 80\\\\ &\text{Maximum value} = 150\\\\ &Q_1=116\\\\ &Q_2=123\\\\ &Q_3=128 \end{aligned} b) The distribution of the data is skewed to the left because the left side of the box plot is longer than the right side of the box plot.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document provides an overview of measures of dispersion, including formulas for range, interquartile range, mean, variance, and standard deviation for grouped data. It includes two examples with solutions:
  
  1. Example 5: Calculation of variance and standard deviation for the volumes of water used daily by a group of families.
  2. Example 6: Construction of a box plot and determination of the distribution shape based on an ogive of starfruit masses.
  
  The document contains questions and answers within the examples.
  
  Label: question-and-answer