Representative values
A long list of numbers is hard to take in at a glance. Often we want a single value that represents the whole set, such as the average score of a class or the most common shoe size sold. Such a value is called a measure of central tendency. The three you need are the mean, the median and the mode.
Arithmetic mean
The mean (average) is the sum of the observations divided by the number of observations.
Mean = sum of all observations ÷ number of observations
Example. Ravi's marks in five tests are 12, 15, 18, 10 and 20. The mean is (12 + 15 + 18 + 10 + 20) ÷ 5 = 75 ÷ 5 = 15.
The mean always lies between the smallest and the largest observation, but it need not be one of the observations.
Range
The range shows how spread out the data is.
Range = highest observation − lowest observation
For Ravi's marks, the range is 20 − 10 = 10.
Mode
The mode is the observation that occurs most often.
Example. Shoe sizes sold in a day: 6, 7, 7, 8, 7, 9, 6, 7, 8. Size 7 occurs four times, more than any other, so the mode is 7. A shop owner cares more about the mode than the mean, because it tells them which size to stock most.
A set can have more than one mode. 2, 3, 3, 5, 5, 6 has two modes, 3 and 5.
Median
The median is the middle value when the data is arranged in order.
Odd number of observations: the median is the middle one. For 3, 8, 5, 11, 7, arrange as 3, 5, 7, 8, 11; the median is 7.
Even number of observations: the median is the mean of the two middle values. For 4, 9, 6, 12, arrange as 4, 6, 9, 12; the median is (6 + 9) ÷ 2 = 7.5.
Choosing the right measure
The mean is affected by extreme values; the median is not.
Example. Five people earn ₹10,000, ₹12,000, ₹11,000, ₹13,000 and ₹2,00,000 a month. The mean is ₹49,200, which is far more than four of the five earn. The median, ₹12,000, represents the group better.
Double bar graphs
A double bar graph shows two sets of data side by side so they can be compared, for example the marks of a class in Term 1 and Term 2 for each subject. The two bars for each category are drawn next to each other in different colours, with a key to explain the colours.
Chance and probability
Some events are certain (the Sun will rise tomorrow), some are impossible (a die shows 7), and many may or may not happen (it will rain tomorrow).
When all outcomes are equally likely:
Probability of an event = number of favourable outcomes ÷ total number of outcomes
Example. A fair die has 6 faces. The chance of getting an even number (2, 4 or 6) is 3/6 = 1/2.
Example. A bag has 3 red and 5 blue balls. The chance of drawing a red ball is 3/8.
A probability is always between 0 (impossible) and 1 (certain).
Summary
Mean = sum ÷ count; range = highest − lowest.
Mode is the most frequent value; median is the middle value of the ordered data.
Extreme values pull the mean but not the median.
Probability = favourable outcomes ÷ total outcomes, between 0 and 1.