Revision and important questions: Tales by Dots and Lines

Tales by Dots and Lines20 min readLesson 6 of 7

6.1 The formula box Mean = sum of values \div number of values. Mean from a frequency table = (sum of value \times frequency) \div total frequency. Median = middle value when ordered; for an even count, average the two middle values. Adding k to every value adds k to the mean; multiplying every value by k multiplies the mean by k. Adding a value above the mean raises it; below the mean lowers it; equal to the mean leaves it unchanged. Spreadsheet: =SUM(B3:G3) for a range, =AVERAGE(B7:D7) for a mean, and D2:D6 names a column range. Line graph: read title, axes, scale and shape; then describe trends, peaks and comparisons that the graph supports.

mean = \frac{\sum x _{i}}{n} median = middle of ordered data

Mean and median

mean_{n e w} = mean_{o l d} + k mean_{n e w} = k \times mean_{o l d}

Scaling and shifting the mean

6.2 Common mistakes Forgetting to order the data before finding the median. For an even count, taking one middle value instead of averaging the two middle values. Adding a value to the data and then dividing by the old count. Assuming the median always changes when a value is added; adding a value far outside the middle may leave it unchanged. Reading a line graph's value from the wrong axis or ignoring the scale. Treating dots in a dot plot as a count of individual days instead of the number of students at that value. Claiming "everyone" or "always" from an average; an average does not describe every individual. Using a frequency table's mean formula without multiplying each value by its frequency.

6.3 Ten exam-style questions with answers Find the mean of 6, 8, 10, 12 and 14. Answer: sum 50, count 5, mean 10. Find the median of 3, 7, 9, 12 and 15. Answer: the middle value is 9. Find the median of 2, 4, 6, 8, 10 and 12. Answer: average of the two middle values (6 + 8) \div 2 = 7. A data set has mean 12 and 8 values. What is its total? Answer: 12 \times 8 = 96. A data set has mean 15 and total 180. How many values does it have? Answer: 180 \div 15 = 12. The mean of five numbers is 9. Four of them are 6, 8, 10 and 12. Find the fifth. Answer: total 45, known sum 36, so the fifth is 9. Every value of a data set is increased by 7. How does the mean change? Answer: the mean also increases by 7. Every value is halved. How does the mean change? Answer: the mean is multiplied by ½. A dot plot has values 4, 4, 5, 6, 6, 6, 8. Find mean and median. Answer: total 39, count 7, mean 39/7 \approx 5.57; median is the fourth value, 6. A line graph shows a shop's sales rising from 20 in January to 35 in May and falling to 25 in July. Describe the trend. Answer: sales rose steadily for four months, peaked at 35 in May, and then fell for two months to 25 in July.

\frac{45 - 36}{1} = 9

Exam question 6: the missing fifth value

\frac{39}{7} \approx 5.57

Exam question 9: mean of a small dot plot

6.4 Reading the formula box The mean formula and the median rule answer different questions, and the two scaling rules explain why the mean is easy to move. The frequency-table formula is the plain mean written in shorthand: each value is repeated as many times as its frequency. The spreadsheet formulas do exactly the same arithmetic, with cell ranges standing in for rows of data. The line-graph instruction is a checklist for observation: title, axes, scale, trend, peak, comparison.

6.5 Exam strategy For a mean question, find the total and the count first; the division is the easy part. For a median question, sort the values before touching the middle. For a table, check whether the frequencies add to the number of values the question implies. For a graph question, quote a specific point from the graph to support each claim. For a data-change question, test the rule on a small example before writing the general statement. When a question gives a mean and a total, check that the two agree; a mismatch usually means the count was misread.

6.6 Checking the revision page Before the final answer, check three things. Does the notation match the data: does a total question use mean \times count, and does a typical value question point to the median? Do the units and labels make sense: heights in centimetres, prices in rupees, time in hours? Does every graph claim name the point or slope that supports it? A short checklist at the end catches most slips, and it also keeps this page focused on the ten questions that matter most. Write the formula first, substitute the numbers second and simplify last; most lost marks come from a swapped middle value or a count taken after a new value was added, not from the arithmetic itself.

in hinglish

Revision ke liye formula box yaad rakho: mean = sum \div count; frequency table mein value \times frequency ka sum \div total frequency; median ke liye data sort karo aur even count par do middle values ka average lo. Har value mein k jodo to mean +k; har value ko k se multiply karo to mean \timesk. Line graph padhne ke liye title, axes, scale, phir trend, peaks aur comparison. Common mistakes : median se pehle sort na karna, even count par ek hi middle value lena, naye value ke baad purane count se divide karna, aur average se "har koi" ya "hamesha" claim karna. Exam mein pehle total aur count nikaalo, data sort karo, aur graph se sirf woh baat kaho jo point/slope se support hoti ho. Do sawal aksar aate hain: mean se total nikaal kar missing value dhoondhna, aur line graph se trend describe karna.