Reading and interpreting data

Data rarely arrives as a tidy list of numbers. You meet it as a table on a sports website, a frequency chart of marks in your report card, a bar graph in a newspaper. Before you can compute a mean or median, you must first read the data correctly.

Idea

Most real data comes in one of three packages:

1. Raw list. Just a string of values. Sort it, then apply mean / median / mode directly.
2. Frequency table. Each value is paired with how often it appears.
3. Grouped or pictorial data. Bar graphs, pictographs , read the heights / counts first.

Frequency tables. Say marks of $10$ students are listed as

This actually means the list $6, 6, 7, 7, 7, 8, 8, 8, 8, 9$. The total count is $\sum f = 10$.

To compute the mean, multiply each value by its frequency and divide by the total: $\text{Mean} = \frac{\sum (f \cdot x)}{\sum f} = \frac{2\cdot 6 + 3\cdot 7 + 4\cdot 8 + 1\cdot 9}{10} = \frac{12+21+32+9}{10} = \frac{74}{10} = 7.4.$

The mode is just the value with the largest frequency , here $8$ (frequency $4$). The median of $10$ sorted values is the average of the $5$th and $6$th values; both are $7$ and $8$ respectively, so median $= \tfrac{7+8}{2} = 7.5$.

Reading bar graphs and pictographs. Treat each bar's height (or each picture's count $\times$ scale) as a frequency. The rules for mean, median and mode are identical to those for a frequency table.

Watch the units. Heights might be in cm or m, prices in $\rupee$ or paise. Convert everything to the same unit before summing.

Always sanity-check. A computed mean should lie between the smallest and the largest value. A median should match the "middle of the sorted list". If your answer is outside the data range, you have made an arithmetic slip.

Worked examples

Example 1. A pictograph shows pencils sold over $5$ days. Each symbol $=$ $10$ pencils. Days have $3, 4, 2, 5, 6$ symbols. Find the mean daily sales.

Convert: $30, 40, 20, 50, 60$. Mean $= \tfrac{30+40+20+50+60}{5} = \tfrac{200}{5} = 40$ pencils per day.

Example 2. Frequency table of family sizes in a survey:

Find the mean family size, mode and median.

$\sum f = 20$. $\sum (f\cdot x) = 2\cdot 4 + 3\cdot 6 + 4\cdot 8 + 5\cdot 2 = 8+18+32+10 = 68$. Mean $= \tfrac{68}{20} = 3.4$. Mode $= 4$ (highest frequency). Median: $20$ sorted values; $10$th and $11$th. Counting: positions $1$–$4$ are $2$; $5$–$10$ are $3$; $11$–$18$ are $4$. So $10$th $= 3$, $11$th $= 4$. Median $= \tfrac{3+4}{2} = 3.5$.

Example 3. Cricket scores of a batter in $6$ innings: $0, 35, 42, 0, 80, 23$. Find mean and median. Comment.

Sort: $0, 0, 23, 35, 42, 80$. Mean $= \tfrac{180}{6} = 30$. Median $= \tfrac{23+35}{2} = 29$. They are close, so $30$ is a fair summary. The two ducks (zeros) lower both equally.

Example 4. Daily rainfall (mm) for a week: $0, 0, 5, 12, 0, 25, 0$. Best central measure?

Mean $= \tfrac{42}{7} = 6$ mm. Median (4th sorted value) $= 0$. Mode $= 0$. Here the mean is the most useful , it captures the total rain spread over the week , even though most days had no rain.

Try it yourself

1. From the table $\{(1,3), (2,5), (3,7), (4,2), (5,3)\}$ (value, frequency), find the mean.
2. Same table , find median and mode.
3. Daily steps walked in a week: $5000, 6500, 7000, 4500, 8000, 7500, 6000$. Mean? Median?
4. A pictograph shows oranges sold over $4$ days; each symbol $=$ $5$ oranges. Counts: $4, 6, 3, 7$. Mean daily sales?
5. Marks of $5$ students: $32, 45, 67, 78, 89$. Sum check: mean should be between $32$ and $89$. Compute and verify.
6. Class survey: number of siblings each student has , $0, 1, 1, 2, 2, 2, 3, 4$. Make a frequency table. Find mean, median, mode.
7. A bar graph shows monthly savings (in $\rupee$): Jan $200$, Feb $250$, Mar $150$, Apr $300$, May $100$. Mean monthly saving?
8. From a frequency table $\{(10,2), (20,3), (30,5)\}$, find $\sum f \cdot x$ and the mean.

Activity

Pick a sports section of a newspaper. Choose one team and one statistic (runs scored / goals / points). Collect the last $10$ games. Make a frequency table. Compute mean, median and mode. Which measure does the commentator usually quote? Why?