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Averages and spread

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Notes de leçon

Measures of Central Tendency

  • Mean = sum of all data values ÷ number of values. It uses every value, so it is sensitive to outliers.
  • Median = middle value when data are ordered from smallest to largest. For an even number of values, it is the mean of the two middle values.
  • Mode = most frequently occurring value. A data set can have more than one mode or no mode.
  • Modal class = the class interval with the highest frequency in a grouped frequency table.
  • Range = largest value − smallest value. It measures spread but can be distorted by outliers.

Averages from a List

  • To find the mean: add all values, then divide by the number of values.
  • To find the median: first order the data, then locate the middle value (or average the two middle values if n is even).
  • To find the mode: identify the value that appears most often.
  • To find the range: subtract the smallest value from the largest.

Averages from a Frequency Table

  • For a frequency table with values x and frequencies f, the mean is \frac{\sum (x \times f)}{\sum f}.
  • The total frequency \sum f is the number of data values, not the number of rows.
  • The median is the value at position \frac{n+1}{2} (or the average of the two middle values if n is even) in the ordered data.
  • The mode is the value with the highest frequency.
  • The range is the difference between the largest and smallest values in the table.

Mean from frequencies

Mean from frequenciesValue xFrequency fx × f1332510326Total1019

Grouped Frequency Tables

  • For grouped data, use the midpoint of each class as a representative value.
  • To estimate the mean: multiply each midpoint by its frequency, sum these products, then divide by the total frequency.
  • The estimated mean is an approximation, not an exact value.
  • The modal class is the class with the highest frequency.
  • The median class is the class that contains the middle value; find it by cumulative frequency.

Estimating a grouped mean

Estimating a grouped meanInteger valuesMidpointfMidpoint × f0–42365–9774910–14121012015–1917585Total—25260

Quartiles and Interquartile Range

  • Quartiles divide ordered data into four equal parts: lower quartile (Q1), median (Q2), upper quartile (Q3).
  • Lower quartile (Q1) = median of the lower half of the data.
  • Upper quartile (Q3) = median of the upper half of the data.
  • Interquartile range (IQR) = Q3 − Q1. It measures the spread of the middle 50% of the data.
  • IQR is generally less sensitive to outliers than the full range. Adding or removing observations can still change quartiles.
  • Worked list examples use medians of the halves, excluding the overall median when n is odd. Follow the convention specified by the question or course.

Quartiles and interquartile range

Quartiles and interquartile range012345678Q1 = 2.5Median = 4.5Q3 = 6.5

Choosing the Most Appropriate Average

  • Use the mean when data are symmetric and have no outliers; it uses all values.
  • Use the median when there are outliers or skewed data; it is resistant to extreme values.
  • Use the mode for categorical data or when the most common value is needed.
  • Justify your choice by referring to the context and the effect of outliers.

Comparing Distributions

  • Compare two data sets using one measure of central tendency (mean or median) and one measure of spread (range or IQR).
  • If outliers are present, use the median and IQR for comparison.
  • If data are symmetric and outlier-free, the mean and range may be appropriate.
  • Always state which average and which spread you are using and explain what they show about the data.

Common Misconceptions

  • When finding the mean from a frequency table, divide by the total frequency, not the number of classes.
  • The median is the middle value, not the middle row of the table.
  • Always order the data before finding the median.
  • The mode is the value with the highest frequency, not the frequency itself.
  • An estimated mean from grouped data is not exact; it is an approximation.
  • The range can be distorted by outliers; consider the IQR instead.

Diapos

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  1. 1.What is the mode of a data set?

    Easy
    • AThe middle value when the data are ordered
    • BThe value that occurs most often
    • CThe arithmetic average of all values
    • DThe difference between the largest and smallest values

  2. 2.Which measure of spread is calculated by subtracting the smallest value from the largest value?

    Easy
    • AInterquartile range
    • BVariance
    • CRange
    • DStandard deviation

  3. 3.To find the median, order the data: use the central value when the count is odd, or the mean of the two central values when the count is even.

    Easy

    True or false?

  4. 4.The heights (in cm) of five students are: 150, 155, 160, 165, 170. What is the median height?

    Medium
    • A155 cm
    • B160 cm
    • C162.5 cm
    • D165 cm

  5. 5.The ages of people in a queue are: 12, 15, 15, 17, 19, 21, 24. What is the interquartile range? Use medians of the lower and upper halves, excluding the overall median when the count is odd.

    Medium
    • A6
    • B7
    • C9
    • D12

  6. 6.Households have 0, 1, 2, 3 and 4 pets with frequencies 3, 5, 4, 2 and 1 respectively. Find the mean number of pets to 1 decimal place.

    Medium
    • A1.2
    • B1.4
    • C1.5
    • D2.0

  7. 7.Which of the following statements about the mean and median are true? (Select all that apply)

    Medium
    • AThe mean is affected by extreme values (outliers).
    • BThe median is always equal to the mean.
    • CThe median is generally less sensitive to extreme values than the mean.
    • DThe mean is always the most appropriate average.
    • EThe median is the middle value when data are ordered.

  8. 8.When estimating the mean from a grouped frequency table, you should divide the total of (midpoint × frequency) by the number of classes.

    Medium

    True or false?

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