{"id":9916,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9916"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"mean-mode-median-graphical-representation-of-frequency-distribution","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/mean-mode-median-graphical-representation-of-frequency-distribution\/","title":{"rendered":"Mean, Mode, Median Graphical Representation Of Frequency Distribution"},"content":{"rendered":"
\n
\n

KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

\n

Unit: <\/strong>Statistics and Probability<\/strong><\/h2>\n

Chapter: <\/strong>Mean, Mode, Median Graphical Representation of Frequency Distribution<\/strong><\/h3>\n

Reference: – Introduction to Statistics, Measures of Central Tendency, Mean (Arithmetic Mean), Median, Mode, Graphical Representation: Bar Graphs, Histograms, Frequency Polygons, Ogives (Cumulative Frequency Curves), Comparison and Application<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • The concepts of mean, median, and mode as measures of central tendency.<\/li>\n
  • How to calculate mean, median, and mode for grouped and ungrouped data.<\/li>\n
  • Various methods of graphical representation of data.<\/li>\n
  • How to interpret and compare different types of graphs.<\/li>\n<\/ul>\n

    Introduction to Statistics<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    Statistics is the branch of mathematics that deals with the collection, organization, analysis, interpretation, and presentation of data. It helps in summarizing and describing the main features of a collection of information.<\/p>\n

    The primary goal is to make sense of data and draw meaningful conclusions from it.<\/p>\n

    [Importance of Statistics]<\/strong><\/p>\n

      \n
    • Used in various fields like economics, business, science, and social sciences.<\/li>\n
    • Helps in decision-making based on data analysis.<\/li>\n
    • Essential for research and forecasting.<\/li>\n
    • Enables comparison between different sets of data.<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      Data Set:<\/strong>\u00a0The test scores of 10 students: 85, 90, 78, 92, 88, 76, 95, 89, 84, 91.
      \nWe can find the average score, the most frequent score, and the middle score<\/p>\n

      [Subtopics]<\/strong><\/p>\n

      1. Types of Data<\/strong><\/p>\n

        \n
      • Ungrouped Data:<\/strong>\u00a0Raw data without any intervals.<\/li>\n
      • Grouped Data:<\/strong>\u00a0Data organized into classes or intervals.<\/li>\n<\/ul>\n

        Key Points:<\/strong><\/p>\n

          \n
        • Frequency:<\/strong>\u00a0The number of times a particular value occurs.<\/li>\n
        • Class Interval:<\/strong>\u00a0A range of values used for grouping data.<\/li>\n
        • Class Mark:<\/strong>\u00a0The midpoint of a class interval.<\/li>\n<\/ul>\n

          Measures of Central Tendency<\/strong><\/p>\n

          [Definition]<\/strong><\/p>\n

          Measures of central tendency are statistical measures that represent the center point or typical value of a dataset. The three main measures are Mean, Median, and Mode.<\/p>\n

          [Importance of Central Tendency]<\/strong><\/p>\n

            \n
          • Provides a single value that represents the entire dataset.<\/li>\n
          • Helps in summarizing large sets of data.<\/li>\n
          • Useful for comparing different datasets.<\/li>\n
          • Forms the basis for more complex statistical analysis.<\/li>\n<\/ul>\n

            Examples<\/strong><\/p>\n

              \n
            • For the data set: 2, 3, 3, 5, 7, the mean is 4, the median is 3, and the mode is 3.<\/li>\n<\/ul>\n

              [Subtopics]<\/strong><\/p>\n

              1. Mean (Arithmetic Mean)<\/strong><\/p>\n

              The mean is the average of all the values in the dataset.<\/p>\n

                \n
              • For Ungrouped Data:<\/strong>\u00a0\n
                \"\"<\/p>\n

                Source: Kapdec.com<\/p>\n<\/div>\n

                , where\u00a0\u2211x<\/em>\u00a0is the sum of all values and n is the number of values.<\/li>\n

              • For Grouped Data:<\/strong>\u00a0\n
                \"\"<\/p>\n

                Source: Kapdec.com<\/p>\n<\/div>\n

                , where\u00a0<\/p>\n

                \"\"<\/p>\n

                Source: Kapdec.com<\/p>\n<\/div>\n

                \u00a0is the frequency of the i-th class and\u00a0<\/p>\n

                \"\"<\/p>\n

                Source: Kapdec.com<\/p>\n<\/div>\n

                \u00a0is the class mark.<\/li>\n<\/ul>\n

                2. Median<\/strong><\/p>\n

                The median is the middle value when the data is arranged in ascending or descending order.<\/p>\n

                  \n
                • For Ungrouped Data:<\/strong>\n
                    \n
                  • If n is odd: Median =\u00a0\n
                    \"\"<\/p>\n

                    Source: Kapdec.com<\/p>\n<\/div>\n

                    \u00a0value.<\/li>\n

                  • If n is even: Median = average of\u00a0\n
                    \"\"<\/p>\n

                    Source: Kapdec.com<\/p>\n<\/div>\n

                    \u00a0and\u00a0<\/p>\n

                    \"\"<\/p>\n

                    Source: Kapdec.com<\/p>\n<\/div>\n

                    \u00a0values.<\/li>\n<\/ul>\n<\/li>\n

                  • For Grouped Data:<\/strong>\u00a0\n
                    \"\"<\/p>\n

                    Source: Kapdec.com<\/p>\n<\/div>\n

                    , where l is the lower limit of the median class, cf is the cumulative frequency of the class preceding the median class, f is the frequency of the median class, and h is the class width.<\/li>\n<\/ul>\n

                    3. Mode<\/strong><\/p>\n

                    The mode is the value that appears most frequently in the dataset.<\/p>\n

                      \n
                    • For Ungrouped Data:<\/strong>\u00a0The value with the highest frequency.<\/li>\n
                    • For Grouped Data:<\/strong>\u00a0\n
                      \"\"<\/p>\n

                      Source: Kapdec.com<\/p>\n<\/div>\n

                      , where l is the lower limit of the modal class,\u00a0<\/p>\n

                      \"\"<\/p>\n

                      Source: Kapdec.com<\/p>\n<\/div>\n

                      \u00a0is the frequency of the modal class,\u00a0<\/p>\n

                      \"\"<\/p>\n

                      Source: Kapdec.com<\/p>\n<\/div>\n

                      \u00a0is the frequency of the class preceding the modal class,\u00a0<\/p>\n

                      \"\"<\/p>\n

                      Source: Kapdec.com<\/p>\n<\/div>\n

                      \u00a0is the frequency of the class succeeding the modal class, and h is the class width.<\/li>\n<\/ul>\n

                      Mean (Arithmetic Mean)<\/strong><\/p>\n

                      [Definition]<\/strong><\/p>\n

                      The arithmetic mean is the sum of all values divided by the number of values. It is the most commonly used measure of central tendency.<\/p>\n

                      [Importance of Mean]<\/strong><\/p>\n

                        \n
                      • Uses all values in the dataset.<\/li>\n
                      • Easy to understand and calculate.<\/li>\n
                      • Suitable for further statistical analysis.<\/li>\n<\/ul>\n

                        Examples<\/strong><\/p>\n

                          \n
                        • Find the mean of the numbers: 10, 20, 30, 40, 50.<\/li>\n<\/ul>\n

                          [Subtopics]<\/strong><\/p>\n

                          1. Calculation for Ungrouped Data<\/strong><\/p>\n

                          \"\"<\/p>\n

                          Source: Kapdec.com<\/p>\n<\/div>\n

                          2. Calculation for Grouped Data<\/strong><\/p>\n

                          Using the formula\u00a0<\/p>\n

                          \"\"<\/p>\n

                          Source: Kapdec.com<\/p>\n<\/div>\n

                          .<\/p>\n

                          Median<\/strong><\/p>\n

                          [Definition]<\/strong><\/p>\n

                          The median is the value that separates the higher half from the lower half of the data set. It is less affected by extreme values (outliers) than the mean.<\/p>\n

                          [Importance of Median]<\/strong><\/p>\n

                            \n
                          • Provides a better measure for skewed distributions.<\/li>\n
                          • Useful when extreme values are present.<\/li>\n
                          • Easy to find for ordinal data.<\/li>\n<\/ul>\n

                            Examples<\/strong><\/p>\n

                              \n
                            • Find the median of: 12, 15, 18, 20, 25.<\/li>\n<\/ul>\n

                              [Subtopics]<\/strong><\/p>\n

                              1. Calculation for Ungrouped Data<\/strong><\/p>\n

                              Arrange in order: 12, 15, 18, 20, 25. n=5 (odd), so Median = 3rd value = 18.<\/p>\n

                              2. Calculation for Grouped Data<\/strong><\/p>\n

                              Using the formula\u00a0<\/p>\n

                              \"\"<\/p>\n

                              Source: Kapdec.com<\/p>\n<\/div>\n

                              .<\/p>\n

                              Mode<\/strong><\/p>\n

                              [Definition]<\/strong><\/p>\n

                              The mode is the value that occurs most frequently in the data set. A dataset may have one mode, more than one mode, or no mode at all.<\/p>\n

                              [Importance of Mode]<\/strong><\/p>\n

                                \n
                              • Useful for categorical data.<\/li>\n
                              • Helps in identifying the most popular or common value.<\/li>\n
                              • Easy to find for nominal data.<\/li>\n<\/ul>\n

                                Examples<\/strong><\/p>\n

                                  \n
                                • Find the mode of: 2, 3, 4, 4, 5, 5, 5, 6, 7.<\/li>\n<\/ul>\n

                                  [Subtopics]<\/strong><\/p>\n

                                  1. Calculation for Ungrouped Data<\/strong><\/p>\n

                                  The value 5 appears three times, so Mode = 5.<\/p>\n

                                  2. Calculation for Grouped Data<\/strong><\/p>\n

                                  Using the formula\u00a0<\/p>\n

                                  \"\"<\/p>\n

                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                  .<\/p>\n

                                  Graphical Representation of Frequency Distribution<\/strong><\/p>\n

                                  [Definition]<\/strong><\/p>\n

                                  Graphical representation involves displaying data in visual forms such as graphs and charts. This makes it easier to understand patterns, trends, and comparisons in the data.<\/p>\n

                                  [Importance of Graphical Representation]<\/strong><\/p>\n

                                    \n
                                  • Provides a quick overview of the data.<\/li>\n
                                  • Helps in identifying patterns and outliers.<\/li>\n
                                  • Makes complex data more understandable.<\/li>\n
                                  • Useful for presentations and reports.<\/li>\n<\/ul>\n

                                    Examples<\/strong><\/p>\n

                                      \n
                                    • Represent the frequency distribution of test scores using a histogram.<\/li>\n<\/ul>\n

                                      [Subtopics]<\/strong><\/p>\n

                                      1. Bar Graph<\/strong><\/p>\n

                                      A graph that uses bars to represent frequencies of different categories. The bars can be vertical or horizontal.<\/p>\n

                                      2. Histogram<\/strong><\/p>\n

                                      A graph that uses bars to represent frequencies of continuous data in class intervals. There are no gaps between the bars.<\/p>\n

                                      3. Frequency Polygon<\/strong><\/p>\n

                                      A line graph formed by joining the midpoints of the tops of the bars in a histogram.<\/p>\n

                                      4. Ogive (Cumulative Frequency Curve)<\/strong><\/p>\n

                                      A graph that represents cumulative frequencies for class intervals. It can be “less than” or “more than” type.<\/p>\n

                                      Comparison and Application<\/strong><\/p>\n

                                      [Definition]<\/strong><\/p>\n

                                      This involves comparing the different measures of central tendency and graphical representations to choose the most appropriate one for a given dataset. It also includes applying these concepts to solve real-world problems.<\/p>\n

                                      [Importance of Comparison and Application]<\/strong><\/p>\n

                                        \n
                                      • Helps in selecting the best measure for a given situation.<\/li>\n
                                      • Enhances critical thinking and analytical skills.<\/li>\n
                                      • Prepares for practical data analysis in various fields.<\/li>\n<\/ul>\n

                                        Examples<\/strong><\/p>\n

                                          \n
                                        • Determine which measure of central tendency is most appropriate for a given dataset.<\/li>\n<\/ul>\n

                                          [Subtopics]<\/strong><\/p>\n

                                          1. When to Use Mean, Median, or Mode<\/strong><\/p>\n

                                            \n
                                          • Mean:<\/strong>\u00a0When data is symmetric and without outliers.<\/li>\n
                                          • Median:<\/strong>\u00a0When data is skewed or has outliers.<\/li>\n
                                          • Mode:<\/strong>\u00a0When dealing with categorical data or identifying the most frequent value.<\/li>\n<\/ul>\n

                                            2. Choosing the Right Graph<\/strong><\/p>\n

                                              \n
                                            • Bar Graph:<\/strong>\u00a0For categorical data.<\/li>\n
                                            • Histogram:<\/strong>\u00a0For continuous data.<\/li>\n
                                            • Frequency Polygon:<\/strong>\u00a0To show trends in continuous data.<\/li>\n
                                            • Ogive:<\/strong>\u00a0To determine medians, quartiles, and percentiles.<\/li>\n<\/ul>\n

                                              [Example: -]<\/strong><\/p>\n

                                              Problem Statement:<\/strong>
                                              \nThe following table shows the distribution of marks obtained by 50 students in a mathematics test.<\/p>\n

                                              \n\n\n\n\n\n
                                              \n

                                              Marks (Class Interval)<\/p>\n<\/td>\n

                                              		\n

                                              0-10<\/p>\n<\/td>\n

                                              		\n

                                              10-20<\/p>\n<\/td>\n

                                              		\n

                                              20-30<\/p>\n<\/td>\n

                                              		\n

                                              30-40<\/p>\n<\/td>\n

                                              		\n

                                              40-50<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

                                              \n

                                              Number of Students (f)<\/p>\n<\/td>\n

                                              		\n

                                              5<\/p>\n<\/td>\n

                                              		\n

                                              10<\/p>\n<\/td>\n

                                              		\n

                                              18<\/p>\n<\/td>\n

                                              		\n

                                              12<\/p>\n<\/td>\n

                                              		\n

                                              5<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                              Source: Kapdec.com<\/p>\n<\/div>\n

                                              a) Find the mean marks.
                                              \nb) Find the median marks.
                                              \nc) Find the modal marks.
                                              \nd) Draw a histogram and frequency polygon for the data.<\/p>\n

                                              Question:<\/strong>\u00a0Solve parts (a) to (c) and describe the construction for (d). Prove your answers by providing a step-by-step solution and giving\u00a0three independent reasons<\/strong>\u00a0supporting your conclusion for part (a) from these domains:\u00a0(A) Direct Formula Application, (B) Assumed Mean Method, (C) Step-Deviation Method.<\/strong><\/p>\n

                                              [Solution: -]<\/strong><\/p>\n

                                              Given:<\/strong>\u00a0Frequency distribution table.<\/p>\n

                                              Step 1: Prepare the table for calculations.<\/strong>
                                              \nWe need class marks (x_i) for mean.<\/p>\n

                                              \n\n\n\n\n\n\n\n\n\n\n
                                              \n

                                              Class Interval<\/p>\n<\/td>\n

                                              		\n

                                              Frequency (f_i)<\/p>\n<\/td>\n

                                              		\n

                                              Class Mark (x_i)<\/p>\n<\/td>\n

                                              		\n

                                              f_i * x_i<\/p>\n<\/td>\n

                                              		\n

                                              Cumulative Frequency (cf)<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

                                              \n

                                              0-10<\/p>\n<\/td>\n

                                              		\n

                                              5<\/p>\n<\/td>\n

                                              		\n

                                              5<\/p>\n<\/td>\n

                                              		\n

                                              25<\/p>\n<\/td>\n

                                              		\n

                                              5<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              10-20<\/p>\n<\/td>\n

                                              		\n

                                              10<\/p>\n<\/td>\n

                                              		\n

                                              15<\/p>\n<\/td>\n

                                              		\n

                                              150<\/p>\n<\/td>\n

                                              		\n

                                              15<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              20-30<\/p>\n<\/td>\n

                                              		\n

                                              18<\/p>\n<\/td>\n

                                              		\n

                                              25<\/p>\n<\/td>\n

                                              		\n

                                              450<\/p>\n<\/td>\n

                                              		\n

                                              33<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              30-40<\/p>\n<\/td>\n

                                              		\n

                                              12<\/p>\n<\/td>\n

                                              		\n

                                              35<\/p>\n<\/td>\n

                                              		\n

                                              420<\/p>\n<\/td>\n

                                              		\n

                                              45<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              40-50<\/p>\n<\/td>\n

                                              		\n

                                              5<\/p>\n<\/td>\n

                                              		\n

                                              45<\/p>\n<\/td>\n

                                              		\n

                                              225<\/p>\n<\/td>\n

                                              		\n

                                              50<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              Total<\/strong><\/p>\n<\/td>\n

                                              		\n

                                              \u03a3f_i = 50<\/strong><\/p>\n<\/td>\n

                                              		\u00a0<\/td>\n\n

                                              \u03a3f_i x_i = 1270<\/strong><\/p>\n<\/td>\n

                                              		\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                              Source: Kapdec.com<\/p>\n<\/div>\n

                                              a) Find the mean marks.<\/strong><\/p>\n

                                              (A) Direct Formula Application<\/strong>
                                              \nMean using direct method:\u00a0<\/p>\n

                                              \"\"<\/p>\n

                                              Source: Kapdec.com<\/p>\n<\/div>\n

                                              <\/p>\n

                                              \"\"<\/p>\n

                                              Source: Kapdec.com<\/p>\n<\/div>\n

                                              So, the mean marks are\u00a025.4<\/strong>.<\/p>\n

                                              (B) Assumed Mean Method<\/strong>
                                              \nLet Assumed Mean (A) = 25.
                                              \nCalculate deviations d_i = x_i – A.<\/p>\n

                                              \n\n\n\n\n\n\n\n\n\n\n
                                              \n

                                              x_i<\/p>\n<\/td>\n

                                              		\n

                                              f_i<\/p>\n<\/td>\n

                                              		\n

                                              d_i = x_i – 25<\/p>\n<\/td>\n

                                              		\n

                                              f_i * d_i<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

                                              \n

                                              5<\/p>\n<\/td>\n

                                              		\n

                                              5<\/p>\n<\/td>\n

                                              		\n

                                              -20<\/p>\n<\/td>\n

                                              		\n

                                              -100<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              15<\/p>\n<\/td>\n

                                              		\n

                                              10<\/p>\n<\/td>\n

                                              		\n

                                              -10<\/p>\n<\/td>\n

                                              		\n

                                              -100<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              25<\/p>\n<\/td>\n

                                              		\n

                                              18<\/p>\n<\/td>\n

                                              		\n

                                              0<\/p>\n<\/td>\n

                                              		\n

                                              0<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              35<\/p>\n<\/td>\n

                                              		\n

                                              12<\/p>\n<\/td>\n

                                              		\n

                                              10<\/p>\n<\/td>\n

                                              		\n

                                              120<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              45<\/p>\n<\/td>\n

                                              		\n

                                              5<\/p>\n<\/td>\n

                                              		\n

                                              20<\/p>\n<\/td>\n

                                              		\n

                                              100<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              Total<\/strong><\/p>\n<\/td>\n

                                              		\u00a0<\/td>\n\u00a0<\/td>\n\n

                                              \u03a3f_i d_i = 20<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                              Source: Kapdec.com<\/p>\n<\/div>\n

                                              Mean =\u00a0<\/p>\n

                                              \"\"<\/p>\n

                                              Source: Kapdec.com<\/p>\n<\/div>\n

                                              \nThis confirms the mean.<\/p>\n

                                              (C) Step-Deviation Method<\/strong>
                                              \nLet Assumed Mean (A) = 25. Class width (h) = 10.
                                              \nCalculate\u00a0<\/p>\n

                                              \"\"<\/p>\n

                                              Source: Kapdec.com<\/p>\n<\/div>\n

                                              .<\/p>\n

                                              \n\n\n\n\n\n\n\n\n\n\n
                                              \n

                                              x_i<\/p>\n<\/td>\n

                                              		\n

                                              f_i<\/p>\n<\/td>\n

                                              		\n

                                              u_i = (x_i – 25)\/10<\/p>\n<\/td>\n

                                              		\n

                                              f_i * u_i<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

                                              \n

                                              5<\/p>\n<\/td>\n

                                              		\n

                                              5<\/p>\n<\/td>\n

                                              		\n

                                              -2<\/p>\n<\/td>\n

                                              		\n

                                              -10<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              15<\/p>\n<\/td>\n

                                              		\n

                                              10<\/p>\n<\/td>\n

                                              		\n

                                              -1<\/p>\n<\/td>\n

                                              		\n

                                              -10<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              25<\/p>\n<\/td>\n

                                              		\n

                                              18<\/p>\n<\/td>\n

                                              		\n

                                              0<\/p>\n<\/td>\n

                                              		\n

                                              0<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              35<\/p>\n<\/td>\n

                                              		\n

                                              12<\/p>\n<\/td>\n

                                              		\n

                                              1<\/p>\n<\/td>\n

                                              		\n

                                              12<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              45<\/p>\n<\/td>\n

                                              		\n

                                              5<\/p>\n<\/td>\n

                                              		\n

                                              2<\/p>\n<\/td>\n

                                              		\n

                                              10<\/p>\n<\/td>\n<\/tr>\n

                                              \n

                                              Total<\/strong><\/p>\n<\/td>\n

                                              		\u00a0<\/td>\n\u00a0<\/td>\n\n

                                              \u03a3f_i u_i = 2<\/strong><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                              Source: Kapdec.com<\/p>\n<\/div>\n

                                              Mean =\u00a0<\/p>\n

                                              \"\"<\/p>\n

                                              Source: Kapdec.com<\/p>\n<\/div>\n

                                              \nThis provides a third verification.<\/p>\n

                                              b) Find the median marks.<\/strong>
                                              \nTotal number of students, n = 50.,\u00a0\u00a0<\/p>\n

                                              \"\"<\/p>\n

                                              Source: Kapdec.com<\/p>\n<\/div>\n

                                              .
                                              \nThe cumulative frequency just greater than or equal to 25 is 33. So, the median class is 20-30.<\/p>\n

                                                \n
                                              • l (lower limit) = 20<\/li>\n
                                              • cf (cumulative frequency of preceding class) = 15<\/li>\n
                                              • f (frequency of median class) = 18<\/li>\n
                                              • h (class width) = 10<\/li>\n<\/ul>\n

                                                Median =\u00a0<\/p>\n

                                                \"\"<\/p>\n

                                                Source: Kapdec.com<\/p>\n<\/div>\n

                                                \n=\u00a0<\/p>\n

                                                \"\"<\/p>\n

                                                Source: Kapdec.com<\/p>\n<\/div>\n

                                                So, the median marks are approximately\u00a025.56<\/strong>.<\/p>\n

                                                c) Find the modal marks.<\/strong>
                                                \nThe class with the highest frequency is 20-30 (f=18). So, the modal class is 20-30.<\/p>\n

                                                  \n
                                                • l = 20<\/li>\n
                                                • f_1 = 18<\/li>\n
                                                • f_0 = 10 (frequency of preceding class)<\/li>\n
                                                • f_2 = 12 (frequency of succeeding class)<\/li>\n
                                                • h = 10<\/li>\n<\/ul>\n

                                                  Mode =\u00a0<\/p>\n

                                                  \"\"<\/p>\n

                                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                                  \n=\u00a0<\/p>\n

                                                  \"\"<\/p>\n

                                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                                  \n=\u00a0<\/p>\n

                                                  \"\"<\/p>\n

                                                  Source: Kapdec.com<\/p>\n<\/div>\n

                                                  So, the modal marks are approximately\u00a025.71<\/strong>.<\/p>\n

                                                  d) Draw a histogram and frequency polygon.<\/strong><\/p>\n

                                                  Histogram:<\/strong><\/p>\n

                                                    \n
                                                  • On the x-axis, take the class intervals (0-10, 10-20, etc.).<\/li>\n
                                                  • On the y-axis, take the frequencies (5, 10, 18, etc.).<\/li>\n
                                                  • Draw bars for each class interval with heights corresponding to their frequencies. Since class widths are equal, the bars will have proportional heights.<\/li>\n<\/ul>\n

                                                    Frequency Polygon:<\/strong><\/p>\n

                                                      \n
                                                    • Find the class marks: 5, 15, 25, 35, 45.<\/li>\n
                                                    • Plot points at (class mark, frequency): (5,5), (15,10), (25,18), (35,12), (45,5).<\/li>\n
                                                    • Join these points with straight lines.<\/li>\n
                                                    • To close the polygon, also plot points at the previous and next class marks with zero frequency: (-5,0) and (55,0), and connect them.<\/li>\n<\/ul>\n

                                                      Final Answers:<\/strong>
                                                      \na) Mean = 25.4
                                                      \nb) Median \u2248 25.56
                                                      \nc) Mode \u2248 25.71
                                                      \nd) Histogram and frequency polygon as described.<\/p>\n

                                                      The mean calculation is rigorously confirmed by three independent methods: Direct, Assumed Mean, and Step-Deviation.<\/p>\n

                                                      \u00a0<\/p>\n

                                                      \u00a0<\/p>\n

                                                      <\/p>\n