Summarising Data

• What is a measure of central tendency?

  Averages represent the 'centre' of data
• What are the three types of averages?

  Mode, median, and mean

  
• What does the mode represent in a data set?

  The most common value in the data
• What is a modal class?

  The class with the highest frequency
• What is the median?

  The middle value of a data set
• How do you find the median of discrete data?

  Order data, find the middle value
• What is the formula for finding the median position?

  ½(n + 1)th value
• If total frequency is 23, what is the median position?

  12th number in the ordered list
• What do you do if the median position is a decimal value?

  Find the two surrounding values and average
• How do you find the median from a frequency table?

  Add frequencies until reaching median position
• What is the median class for grouped data?

  The class interval containing the median position
• Why is the median for grouped data always an estimate?

  Exact values are not known in groups
• What is the first step to estimate the median using linear interpolation?

  Use ½ n to find the median position
• What does cumulative frequency (CF) help identify?

  The class interval containing the median
• How do you find the median's position in the group?

  Subtract CF of the group above from ½ nth value
• What is the formula for the mean?

  $\bar{x} =$$\frac{\sum x}{n}$
• What does $\sum$ represent in the mean formula?


  Sum of all values
• How do you calculate the mean for discrete data?

  Add all values and divide by number of values
• What is the first step in calculating the mean from a frequency table?

  Add an extra column labeled $f \times x$
• How do you find the total in a frequency table?

  Add up the $f \times x$ column
• What is the formula for the mean from a frequency table?

  $\frac{\sum f x}{\sum f}$
• What is the first step in calculating the mean from a grouped frequency table?

  Add two extra columns for midpoint and $f \times midpoint$
• How do you calculate the midpoint of class intervals?

  Average the upper and lower bounds of the class
• What is the formula for the weighted mean?

  $\frac{\sum(weight \times value)}{\sum weights}$
• When is the geometric mean useful?

  For comparing different growth rates
• What is the formula for the geometric mean?

  $\sqrt[value_1 \times value_2 \times \ldots \times value_n]{n}$
• Why might you transform data?

  To simplify calculations with large numbers
• What is the first step in transforming data?

  Add or subtract a number from data values
• What is the effect of adding a value greater than the median?

  The median might increase
• How does removing a value greater than the median affect it?

  The median might decrease
• What happens to the mean if you add a value greater than it?

  The mean increases
• What are the advantages and disadvantages of using mode, median, and mean?

  Advantages:

  • Mode: Easy to use, unaffected by extremes
  • Median: Unaffected by outliers, best for skewed data
  • Mean: Uses all data, can calculate standard deviation

  
  

  Disadvantages:

  • Mode: May not exist, not representative
  • Median: May not be a data value, not always representative
  • Mean: Affected by extremes, may not be a data value
• What is the range in statistics?

  The difference between largest and smallest values
• What is the formula for range?

  $Range =$$Largest Value - Smallest Value$
• What does the interquartile range (IQR) represent?

  The middle 50% of the data
• What is the formula for IQR?

  $IQR =$$Upper Quartile - Lower Quartile$
• How do you find the lower quartile (LQ) for discrete data?

  $LQ =$$\frac{1}{4}(n + 1)th value$
• How do you find the upper quartile (UQ) for discrete data?

  $UQ =$$\frac{3}{4}(n + 1)th value$
• How do you calculate IQR for discrete data?

  $IQR =$$UQ - LQ$
• How do you find LQ and UQ for grouped data?

  Use cumulative frequency curve to find positions