{"id":287,"date":"2020-06-29T14:19:28","date_gmt":"2020-06-29T18:19:28","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=287"},"modified":"2026-02-18T17:38:23","modified_gmt":"2026-02-18T22:38:23","slug":"2-4-five-number-summary-and-boxplot","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/2-4-five-number-summary-and-boxplot\/","title":{"raw":"2.4 Five-Number Summary and Boxplot","rendered":"2.4 Five-Number Summary and Boxplot"},"content":{"raw":"The five-number summary of a data set consists of the minimum (the smallest observation), [latex]Q_1, Q_2,Q_3[\/latex] and the maximum (the largest observation).\r\n\r\nThese five numbers together give us a brief idea about the distribution of the data: [latex]Q_2[\/latex]\u00a0(the median) is the centre of the distribution, the range (the difference between the maximum and the minimum) and the IQR (the difference between [latex]Q_3[\/latex]\u00a0and [latex]Q_1[\/latex]) tell us the spread (variation) of the data. The difference between [latex]Q_1[\/latex]\u00a0and the minimum, between [latex]Q_2[\/latex]\u00a0and [latex]Q_1[\/latex], between [latex]Q_3[\/latex]\u00a0and [latex]Q_2[\/latex], and between the maximum and [latex]Q_3[\/latex]\u00a0give the range of the first, second, third and fourth 25% of the data respectively. Moreover, the five-number summary helps us identify outliers, those observations that are far away from the bulk of the data.\r\n

2.4.1 Identify Outliers <\/strong><\/h2>\r\nOutliers are observations far away from the majority of the data. Quantitatively, any observation that falls outside the interval of (lower limit, upper limit) is considered as an outlier. The upper and lower limits are defined as:\r\n\r\n[latex] \\text{lower limit} = Q_1 - 1.5 \\times IQR; \\quad \\text{upper limit} = Q_3 + 1.5 \\times IQR.[\/latex]\r\n
\r\n

Example: Identify Outliers<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nIdentify the outliers for the data 3, 1, 9, 7, 5, 11, 21 if any.\r\n\r\nSteps:\r\n
    \r\n \t
  1. Find the quartiles. Refer to Example 4, part (a), [latex]Q_1 = 4, Q_2=7, Q_3=10[\/latex].<\/li>\r\n \t
  2. [latex]IQR = Q_3 - Q_1 = 10-4=6[\/latex]<\/li>\r\n \t
  3. [latex]\\text{lower limit}=Q_1 -1.5 \\times IQR=4-1.5 \\times 6=-5[\/latex]<\/li>\r\n \t
  4. [latex]\\text{upper limit}=Q_3+1.5 \\times IQR=10+1.5 \\times 6=19[\/latex]<\/li>\r\n<\/ol>\r\nSince 21 > 19, it is outside the interval (-5, 19), 21 is an outlier.\r\n\r\n<\/div>\r\n<\/div>\r\n
    \r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    \r\n

    Exercise: Choose Proper Measures<\/p>\r\n\r\n<\/header>\r\n

    \r\n\r\nBased on the histogram and five-number summary of the data, answer the following questions.<\/a>\r\n

    Table 2.3<\/strong>: Five-Number Summary of the Data<\/a><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    \r\n
    Summary<\/strong><\/div><\/td>\r\n
    		\r\n
    Min<\/strong><\/div><\/td>\r\n
    		\r\n
    Q1<\/sub><\/strong><\/div><\/td>\r\n
    		\r\n
    Median<\/strong><\/div><\/td>\r\n
    		\r\n
    Q3<\/sub><\/strong><\/div><\/td>\r\n
    		\r\n
    Max<\/strong><\/div><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
    \r\n
    \u00a0<\/strong><\/div><\/td>\r\n
    		\r\n
    0.1<\/div><\/td>\r\n
    		\r\n
    2<\/div><\/td>\r\n
    		\r\n
    3.5<\/div><\/td>\r\n
    		\r\n
    5<\/div><\/td>\r\n
    		\r\n
    32<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"attachment_255\" align=\"aligncenter\" width=\"300\"]\"Histogram Figure 2.2<\/strong>: Histogram of the Data [Image Description (See Appendix D Figure 2.2)<\/a>][\/caption]\r\n
      \r\n \t
    1. Comment on the distribution (shape, centre, spread).<\/li>\r\n \t
    2. Are there any outliers in the data?<\/li>\r\n \t
    3. Provide proper measures of the centre and spread of the data. Explain why.<\/li>\r\n<\/ol>\r\n
      Show\/Hide Answer<\/summary>\r\n
        \r\n \t
      1. Comment on the distribution (shape, centre, spread).\r\nThe distribution is unimodal, skewed to the right with a median 3.5 and [latex]IQR = 5-2=3[\/latex].<\/span><\/li>\r\n<\/ol>\r\n
          \r\n \t
        1. Are there any outliers in the data?\r\nYes. [latex]\\text{Upper limit} = Q_3 + 1.5 \\times IQR = 5 + 1.5 \\times 3 = 9.5[\/latex].<\/span>\r\n<\/span>Any observation greater than 9.5 is an outlier.<\/span><\/li>\r\n<\/ol>\r\n
            \r\n \t
          1. Provide proper measures of the centre and spread of the data. Explain why.\r\nUse median for the centre and IQR for the spread due to outliers and strong skewness.<\/span><\/li>\r\n<\/ol>\r\n<\/details><\/div>\r\n<\/div>\r\n

            2.4.2 Boxplot <\/strong><\/h2>\r\nA boxplot<\/strong>, also called a box-and-whisker plot, is a useful tool to display the centre and spread of a data set by providing a graphical representation of the five-number summary as well as potential outliers. Steps to draw a boxplot:\r\n
              \r\n \t
            1. Calculate the five-number summary: minimum, [latex]Q_1, Q_2, Q_3[\/latex], and maximum.<\/li>\r\n \t
            2. Calculate the lower and upper limits: [latex]\\text{lower limit}=Q_1 -1.5 \\times IQR[\/latex], and [latex]\\text{upper limit} = Q_3 + 1.5 \\times IQR.[\/latex]<\/li>\r\n \t
            3. Find the adjacent values<\/strong>, the largest and smallest observations within the lower and upper limits<\/strong>. Identify the potential outliers (observations beyond the upper and lower limits), if any exist.<\/li>\r\n \t
            4. Draw short horizontal lines at [latex]Q_1, Q_2, Q_3[\/latex] , and connect them with vertical lines to form a box.<\/li>\r\n \t
            5. Draw very short horizontal lines at the adjacent values and then draw the whiskers by connecting the adjacent values and the box with vertical lines.<\/li>\r\n \t
            6. Plot each potential outlier with an asterisk.<\/li>\r\n \t
            7. Put labels and the title.<\/li>\r\n<\/ol>\r\n
              \r\n\r\n\"\"\r\n\r\n<\/div>\r\n