13. Statistics

8 topics2 h

Chapter 13 of NCERT Mathematics for Class 10: Statistics. Three sections — Mean of grouped data, Mode of grouped data and Median of grouped data. Eight videos, 2 h in all.

Worked answers to the 22 exercise questions in this chapter: Exercise 13.1 · Exercise 13.2 · Exercise 13.3

0 of 8 done — sign in to track this chapter

Part 1

Mean of grouped data

13 minGrouping loses the raw values, so we stand the class mark in for themThirty students, one paper. Work out the average and you get 59.3. Sort the same thirty marks into six classes, work out the average from that table, and you get 62. Neither number is a mistake, and the 2.7 between them is exactly the amount by which the classes fail to balance about their middles.13 min · 13 parts15 minThe direct method, and where it becomes unwieldyThe direct method is not a method. Write the formula for the mean of a table of values above the formula for the mean of a table of classes and they are the same characters in the same order - only the x column has changed hands. That is why this one route to a grouped mean needs no proof, and why the two that save you arithmetic do.15 min · 13 parts14 minShifting the origin: why guessing a centre cannot change the answerSubtract any number you like from every class mark, average what is left, and the result falls short of the true mean by exactly that number. Never by more, never by less. So the centre you measure from is a genuinely free choice, and a bad guess costs you arithmetic — not a single decimal of accuracy.14 min · 13 parts15 minDividing through by the class width to shrink the arithmetic furtherThe step in the step-deviation method is usually introduced as the class width, and on a table of six classes with three different widths that description stops making sense. The method runs anyway, at a step of 20 that matches none of them, and lands on exactly the number the direct method lands on - because the algebra never asked h to be the class width. It only ever asked h not to be nought.15 min · 13 parts

Part 2

Mode of grouped data

14 minFinding the busiest class, then placing the mode inside itFinding the busiest class is half of finding the mode, and it is the half everybody remembers. The other half is deciding where inside that class the mode actually goes - and that is settled by the two classes on either side, not by the busy one. Draw two lines corner to corner across three bars and they cross at the answer, leaning toward whichever neighbour is more crowded.14 min · 13 parts

Part 3

Median of grouped data

15 minRunning totals, and converting between the two cumulative tablesA frequency table answers 'how many are here?' and cannot answer 'which class is the 27th script in?'. A running total answers the second, and it can be run in either direction - so the same distribution produces two different-looking columns whose entries add to the total at every shared boundary. That is not a rule of arithmetic; it is what a clean split means.15 min · 13 parts15 minLocating the middle class and interpolating across itGrouping throws the actual values away, so the middle observation still exists but the table cannot name it. The median formula is not five symbols to memorise - it is one sentence: find the class the middle falls in, assume its observations are evenly spread, go as far into that class as the middle stands through its count, and measure that in the data's own units.15 min · 13 parts15 minWhich of the three averages a given question actually wantsThirty scripts give a mean of 62, a median of 62.5 and a mode of 52, and nobody has made an arithmetic slip. The three are not three attempts at one quantity - they are exact answers to three different questions, and the whole difference between them is how much of the table each one actually reads.15 min · 13 parts

What you will be able to do

  • Compute the mean of an ungrouped frequency distribution as Σf·x ÷ Σf, laying the products out in a column
  • State the direct-method formula and identify each symbol in a given table
  • Choose an assumed mean from a table and justify the choice on arithmetic grounds
  • Build the u column as (x − a) ÷ h for a given assumed mean and step
  • Find the mode of an ungrouped data set by tabulating frequencies
  • Build a cumulative frequency column from a frequency column, upward and downward
  • Compute n ÷ 2 and identify the median class from a cumulative frequency column
  • Compute all three measures for one grouped distribution and set them side by side

What this chapter assumes you already know

  • The mean of ungrouped data as total-of-values divided by number-of-values
  • Reading a frequency table: that a frequency f against a value x means x was observed f times, so it contributes f copies to the total
  • Sigma notation for a sum, and that Σf is the number of observations while Σfx is the sum of them
  • Forming a grouped frequency distribution from a list of raw values, from Class IX
  • Upper and lower limits of a class, and the arithmetic average of two numbers
  • Reading a grouped frequency table and writing down class marks
  • Multiplying a two- or three-digit number by a two- or three-digit number, and totalling a long column
  • Division producing a non-terminating decimal, and rounding a result to two decimal places

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "One of 59.3 and 62 is a mistake."
  • "Grouping always pushes the mean up."
  • "The class mark is the average of the observations in the class."
  • "A mark of 40 could go in either 25–40 or 40–55."
  • "Wider classes are simpler, so use them."
  • "Σf·x ÷ Σf is a new formula for grouped data."
  • "The direct method is the crude one and the others are better."
  • "Choose the method by the size of the answer."