Mean and median from a frequency table, by hand and in a spreadsheet

What to assume they know

• Mean = total ÷ count, and the median as the middle value or the average of the two middle values — The mean as the point where the distances balance, Whether adding a value raises or lowers the median
• Reading a frequency table, and reading a dot plot as a frequency table with the counts drawn as stacks
• Multiplying a small whole number by a small whole number, and adding a column of eight products
• Dividing to two decimal places, e.g. 188 ÷ 36
• Counting positions in a sorted list
• The idea of an address made from a letter and a number, as on a grid or a map

What they should be able to do

• Explain what a frequency table records, and reconstruct the underlying list from one
• Say why averaging the distinct values of a frequency table is wrong, and what quantity that computation actually gives
• Compute a mean from a frequency table as a weighted total over the total frequency
• Identify the correct divisor for such a mean, and say why it is not the number of rows
• Locate the median from a frequency table by accumulating frequencies, and state which positions a given value occupies
• Report a mean that does not come out exactly, and say what has been rounded
• Name a spreadsheet cell from its column letter and row number, and read a value out of a named cell
• Write a range as a start cell and an end cell, and say which values it collects
• Write a formula that totals a row and one that averages part of a row, and predict its result before the sheet computes it
• Compute the mean, median, smallest and largest value of a data set given as a frequency table

Where it usually goes wrong

• "Average the numbers in the left-hand column." This is printed as the tempting answer for a reason. It computes the average of the reported values, which is a different question, and it is wrong by more than a rounding: 6.5 against 5.22.
• "Divide by the number of rows." Divide by the total frequency. Eight rows, thirty-six students. The dart table makes it starker: ten rows, sixty-two students, two of the rows contributing nobody at all.
• "A frequency of 1 and a frequency of 11 are both just one row, so they count the same." That is the whole error, stated plainly. Show the table unfolded into thirty-six tick marks once and the objection disappears.
• "You must write out all the values to find the median." The chapter asks this question directly and answers it: accumulate the frequencies instead. For thirty-six values writing them out is merely tedious; for the sixty-two dart throwers it is a real obstacle.
• "The median is the middle row of the table." The middle row here is between 5 and 6, and the median is 5. Rows are values, not positions.
• "5.22 is the family size." No family has 5.22 members. It is the balance point of thirty-six family sizes, rounded to two places from 5.2222…
• "A spreadsheet knows statistics." It knows how to add up a named block. Every formula in this section is one of the two operations already done by hand, addressed differently.
• "B7 is a column." The chapter's own question says column and means cell. Use the slip: a letter alone names a column, a number alone names a row, and it takes both to name a cell.
• "Above 30 includes 30." Ashwin's Social Science mark is exactly 30. Strict comparisons are half the work in reading a table, and this item is built on one.

Questions to check understanding

• Compute the mean from a frequency table, showing the products and the total frequency
• Say what the average of a frequency table's distinct values represents, and why it is not the mean asked for
• Locate the median from a frequency table using running totals, and name the positions a given value occupies
• Report the smallest, largest, mean and median of data given as a frequency table
• Predict what a stated spreadsheet formula will return, given the table
• Write the formula for the average of a named column, given where the data begins and ends
• Name the cell holding a stated entry, and read out the entry in a named cell
• Convert a dot plot into a frequency table and back
• Recompute a mean and a median after every value gains one, using the shift rule rather than starting again

Examples worth working on the board

Values marked printed appear on the page. Values marked not in the book are worked out here on the chapter's stated inputs. The chapter prints no answers to any exercise item and Part II has no answer appendix.

• The family-size table (Part II p.110, printed, two columns headed Number and Frequency): 3 with frequency 3; 4 with 11; 5 with 9; 6 with 7; 7 with 3; 8 with 1; 9 with 1; 10 with 1. Eight rows, and not in the book: thirty-six students in all.
• The tempting wrong route (Part II p.110, printed as the answer someone in the class might give): 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 52, and 52 ÷ 8 = 6.5. Not in the book: that is the average of the eight distinct family sizes reported, which is a real quantity and not the one asked for. Naming what the wrong computation computes is more useful than calling it wrong.
• The weighted computation (Part II p.110, printed in full): (3 × 3) + (4 × 11) + (5 × 9) + (6 × 7) + (7 × 3) + (8 × 1) + (9 × 1) + (10 × 1), over 3 + 11 + 9 + 7 + 3 + 1 + 1 + 1. The products are printed as 9, 44, 45, 42, 21, 8, 9, 10, the total as 188, the divisor as 36, and the answer as 5.22.
• What 5.22 is (not in the book). 188 ÷ 36 is 5.2222…, so the printed 5.22 is rounded. Worth one sentence saying so: an average family size that is not a whole number is not an error, and neither is a rounded report — but a class should know which of the two they are looking at.
• The median from the same table (Part II p.110, printed). With 36 values the median is the average of the eighteenth and nineteenth. The page accumulates: 3 + 11 = 14, so the fourteenth value in sorted order is 4; 3 + 11 + 9 = 23, so the twenty-third is 5; therefore every position from the fifteenth to the twenty-third holds 5, and the median is 5.
• Why that is the whole method (not in the book). The running totals are the last position each value occupies. Once you can name the last position of a value, you can answer "what sits at position 18?" for any position at all, which is exactly what a median needs and exactly what writing out thirty-six numbers would have given you more slowly.
• The frequency picture (Part II p.110, printed as a small chart to the right of the median discussion, vertical axis 0 to 12 in twos, horizontal axis 2 to 11). Each family size carries a column of dots as tall as its frequency, so the table and the plot are the same object drawn twice. Read on the printed page; none of it extracts.
• The marks table (Part II p.111, printed in full): twenty-two students against six subjects — Odia, Telugu, English, Maths, Social Science, Science. The names in printed order are Ratna, Nagesh, Ashwin, Farooq, Mrinal, Gowri, Pankaj, Jaya, Ganesh, Shravan, Aishwarya, Hari, Trupti, Veeresh, Vidhya, Sanskruti, Shanker, Vyshnavi, Govind, Shiva, Tarun and Jyothi. All 132 marks are printed; the ones the rest of this brief needs are Nagesh (41, 43, 48, 39, 40, 39), Farooq (47, 46, 38, 42, 49, 44), Ashwin (29, 31, 33, 34, 30, 28) and Gowri (27, 29, 34, 31, 32, 30).
• The spreadsheet snapshot (Part II p.112, printed as a screenshot; read on the printed page because none of its lettering extracts). Column A is headed Name, then B Odia(R1), C Telugu(R2), D English(R3), E Maths, F Social, G Science, H Total. Row 1 holds the headings and rows 2 to 8 hold the first seven students, Ratna through Pankaj. Column H is empty. Note two differences from the printed table: the sheet abbreviates Social Science to Social, and it tags the three languages R1, R2 and R3.
• The addressing questions (Part II p.112, printed). Farooq's Maths mark is in E5, which the page states. Two more are asked and not answered: what sits in B7, and in which subjects Ashwin scored above 30. Not in the book: B7 holds Gowri's Odia mark, 27 — and the question is worth pausing on because it calls B7 a column when it is a cell. Ashwin cleared 30 in Telugu, English and Maths; his Social Science mark is exactly 30, which does not qualify, and that boundary is the point of the item.
• The two printed formulas (Part II p.112, printed): =SUM(B3:G3) totals Nagesh's six marks and =AVERAGE(B7:D7) averages Gowri's first three. The page also gives B3:G3, B7:D7 and D2:D6 as examples of the notation, the last being one subject's marks for the first five students. Not in the book: the two results are 250 and 30 exactly — good numbers to predict aloud before the sheet answers, and 250 is corroborated by the formula-bar preview visible on Part II p.113.
• The three open spreadsheet tasks (Part II p.113, printed, no answers): write the formula for the class average in Science; find out whether the class average in Odia beats the class average in Telugu; put the subject averages in a row of their own below the data; and total each student's marks. Not in the book: with the twenty-two students in rows 2 to 23, the Science average is =AVERAGE(G2:G23); the Odia marks total 687 for an average of about 31.2 and the Telugu marks total 739 for about 33.6, so the answer to the comparison is no. Both totals are worked out here on the printed table.
• The QR code and the Note to the Teacher (Part II p.113, printed). The page carries a QR code offering the table as a downloadable file, and a Note to the Teacher saying any spreadsheet application will do, that students can share a machine in groups, and that one screen can be projected for the class if that is all there is.
• Item 10 (Part II pp.115–116, printed as a dot plot on an axis 0 to 10, with the hint in the question that four students rode twice). Counts read off the printed page, one per half of the plot: 0 rides — 3 students; 1 — 1; 2 — 4; 3 — 7; 4 — 7; 5 — 5; 6 — 4; 7 — 6; 8 — 3; 9 — none; 10 — 2. Not in the book: 42 students, 193 rides, mean 4.60 and median 4. Of the four claims offered: not everyone rode at least once, since three students are at zero; and some students must have ridden more than once on some day, because a week holds seven days and eight or ten rides cannot fit one a day, so option (c) is supported. Option (d), "exactly 5 students", is not a valid inference. The seven-day argument forces at least five — the three at eight rides and the two at ten — but the plot records weekly totals only, so a student sitting at 3, 4, 5, 6 or 7 rides may also have doubled up on some day, and nothing on the plot excludes it. The data forces five and permits more. This is worth dwelling on rather than glossing: the whole of §5.2, and this module's thesis, is the difference between supported, contradicted and not-addressed, so teaching "exactly five" here teaches the very error the chapter was built to correct. Part (e) adds one ride to everybody, which by the shift rule of Which added values move the mean, and in which direction gives mean 5.60 and median 5.
• Item 11 (Part II p.116, printed as a two-row table). Throws needed to hit the centre, 1 through 10, against the number of students: 1, 0, 0, 1, 4, 9, 12, 15, 10, 10. Not in the book: 62 students, 473 throws in all, mean about 7.63, median 8, smallest 1 and largest 10. The two zero frequencies are the useful detail — a value with frequency zero contributes nothing to the total and nothing to the accumulation, which is the cleanest possible demonstration that it is the frequencies and not the rows that matter.

Figures to have open

• The frequency table and its unfolded form, side by side. The chapter prints the table (Part II p.110); the unfolding is added here and it is the figure the whole topic turns on.
• A position strip from 1 to 36 with each value's block shaded and its last position labelled. Not in the chapter, and it is what makes the accumulation method visible rather than procedural.
• The frequency plot from Part II p.110, redrawn — the table and the picture together are worth one screen.
• A spreadsheet grid with column letters, row numbers, the marks of a handful of students, one cell highlighted with its address, and one range highlighted with its Start:End label. Redraw rather than reproduce the printed screenshot; keep column H empty so the totals can be filled in.
• The cycle-ride dot plot (Part II p.115) and the dart table (Part II p.116), both redrawn.
• No photograph is needed. The dartboard illustration on Part II p.116 is decorative.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 5, "Tales by Dots and Lines", §5.1 "The Balancing Act", Part II pp.110–113. Two unnumbered bold subheadings cover this stretch and can be named but not cited by number: "Mean and Median with Frequencies" (Part II p.110) and "Spreadsheets" (Part II pp.111–113). The marks table fills Part II p.111 by itself; the snapshot, the addressing questions and the two formulas are on Part II p.112; the formula bar, the QR code and the Note to the Teacher are on Part II p.113, immediately above the chapter's first Figure it Out.
• Exercise items owned by this topic: Part II pp.115–116 item 10 (all parts, with part (e) leaning on Which added values move the mean, and in which direction) and Part II p.116 item 11.
• Sibling topics: the mean's and median's definitions and their response to change are The mean as the point where the distances balance, Which added values move the mean, and in which direction and Whether adding a value raises or lowers the median; recovering an unknown from a stated average is Working backwards from an average to a missing value.
• Forward pointer: §5.2 begins on Part II p.116 directly below item 11 and is covered by the four m03 topics.