Pariksha SaathiLesson 8 of 8 · Mathematics, Classes VI-VIII (Paper II)
Data Handling: Mean, Median, Mode, Graphs and Probability
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Collecting and organising data
Data recorded as it comes is raw data; a frequency table with tally marks (the fifth mark crossing the other four) shows how often each value occurs, and the range is the highest value minus the lowest. Large data are grouped into class intervals of equal width, such as 10-20, 20-30 and 30-40, where 10 is the lower limit, 20 the upper limit and 10 the class size; by convention a value equal to an upper limit, such as 20, is counted in the next interval, 20-30. A pictograph shows data with symbols and a key, so if one symbol stands for 10 cars, half a symbol stands for 5.
Mean, median and mode
These representative values describe a whole data set by one number. The arithmetic mean is the sum of the observations divided by their number; the median is the middle value once the data are arranged in order, and for an even number of values the mean of the middle two; the mode is the value that occurs most often. For the marks 42, 37, 50, 45, 37, 48 and 56 the sum is 315, so the mean is 315 ÷ 7 = 45; in order they read 37, 37, 42, 45, 48, 50, 56, so the median is 45; the mode is 37; and the range is 56 − 37 = 19. For 10, 14, 18, 20, 25 and 31 the median is (18 + 20) ÷ 2 = 19. A data set can have more than one mode, or none if no value repeats.
Using the measures
The mean always lies between the smallest and largest values. Adding 5 to every value raises the mean, median and mode by 5, and doubling every value doubles them. A missing value comes from the total: if the mean of five numbers is 18 and four of them are 15, 20, 12 and 25, the fifth is 90 − 72 = 18; if the mean of six numbers is 20 and falls to 18 when one is removed, the removed number is 120 − 90 = 30. Extreme values pull the mean, so the median better represents the incomes of a village with one very rich family, while the mode answers which size or colour is most common, as a shopkeeper ordering shoes needs to know.
Bar graphs and histograms
A bar graph uses bars of equal width with equal gaps between them, their heights showing values on a stated scale (1 unit = 10 pupils); a double bar graph places two bars side by side to compare two sets, such as a class's marks in two terms. A histogram shows grouped data: the class intervals lie along the horizontal axis, the bars touch with no gaps, each bar's width is the class size and its height the frequency. When the first interval does not start at 0, a zigzag (kink) on the horizontal axis shows the break in the scale.
Pie charts
A pie chart, or circle graph, shows how a whole is divided into parts; each sector's central angle is the same fraction of 360° as the part is of the whole, so central angle = (value ÷ total) × 360°, and a percentage becomes degrees when multiplied by 3.6. If 36 pupils choose cricket (15), football (9), hockey (6) and kabaddi (6), the angles are 150°, 90°, 60° and 60°, adding up to 360°. Read backwards, a sector of 72° in a monthly expenditure of ₹50,000 is 72/360 = 1/5 of it, that is ₹10,000, and a 25% share is a 90° sector.
Chance and probability
An experiment such as tossing a coin or throwing a die has outcomes, and when they are equally likely the probability of an event is the number of outcomes favourable to it divided by the total number of outcomes. A die has 6 outcomes, so P(even number) = 3/6 = 1/2, P(number greater than 4) = 2/6 = 1/3 and P(prime) = 3/6 = 1/2, the primes being 2, 3 and 5. A bag of 3 red, 5 blue and 2 green balls gives P(blue) = 5/10 = 1/2 and P(not red) = 7/10. A probability lies between 0 for an impossible event and 1 for a sure event, and P(not E) = 1 − P(E). Tossing two coins gives the four outcomes HH, HT, TH and TT, so P(exactly one head) = 2/4 = 1/2.
In the classroom
Pupils find the median without first arranging the data, give the highest frequency as the mode instead of the value that has it, leave gaps between histogram bars, and use percentages directly as degrees in a pie chart. Collecting the class's own data (heights, birthdays, ways of coming to school), drawing graphs from it and asking what each graph does and does not show builds the habit of interpreting data, not only computing with it; tossing a coin 50 times shows that the proportion of heads comes close to 1/2 but need not equal it.
Key facts
- Marks 42, 37, 50, 45, 37, 48, 56: mean 315 ÷ 7 = 45, median 45, mode 37, range 19.
- Even number of values: the median of 10, 14, 18, 20, 25, 31 is (18 + 20) ÷ 2 = 19.
- Missing value: the mean of 6 numbers is 20 and of the remaining 5 is 18, so the removed number is 120 − 90 = 30.
- Adding a constant to every value adds it to the mean, median and mode; multiplying every value by a number multiplies them by it.
- Class intervals 10-20, 20-30: the value 20 is counted in 20-30; the class size is 10.
- A histogram's bars touch and show grouped data; a bar graph has equal gaps between bars.
- Pie chart: central angle = (value ÷ total) × 360°, so 15 of 36 pupils is 150°, and 25% is 90°.
- Die: P(even) = 1/2, P(greater than 4) = 1/3, P(prime) = 1/2; for any event P(not E) = 1 − P(E).
- A bag of 3 red, 5 blue and 2 green balls: P(blue) = 5/10 = 1/2 and P(not red) = 7/10.
- Choose the median when there are extreme values, the mode for the most common size or choice, and the mean for an overall average.
Quick revision
Range = highest − lowest; value on an upper class limit goes to the next interval; mean = sum ÷ number; median = middle value after ordering (mean of middle two if even); mode = most frequent value; adding k to every value adds k to the mean, median and mode; bar graph has gaps, histogram none; pie-chart angle = value ÷ total × 360°, 1% = 3.6°; probability = favourable ÷ total equally likely outcomes, from 0 to 1; P(not E) = 1 − P(E).