Basic Statistics and Number Series: Methods and Worked Examples

Measures of central tendency

For ungrouped data (a plain list of values) three averages are examined.

• Median position for n values in order: the (n + 1)/2 th value. If n is odd this is a single value; if n is even, take the mean of the two middle values.
• A data set can have one mode, two modes (bimodal) or no mode.
• From the mean: Sum = Mean × n. This is used to find a missing value or to adjust for a value added or removed.

Range

Range = Largest value - Smallest value. It is the simplest measure of spread.

Simple probability

When all outcomes are equally likely:

P(E) = Number of favourable outcomes ÷ Total number of possible outcomes

• A probability lies between 0 and 1.
• P(not E) = 1 - P(E).
• One fair die has 6 outcomes; two dice have 6 × 6 = 36 ordered outcomes; two coins have 4.
• The prime numbers on a die are 2, 3 and 5. The number 1 is not prime.

Number series

Find the rule that links each term to the next. Check the patterns in this order:

1. Constant difference (arithmetic): 7, 11, 15, 19 adds 4 each time.
2. Constant ratio (geometric): 3, 6, 12, 24 doubles each time.
3. Growing differences: 2, 6, 12, 20, 30 has differences 4, 6, 8, 10. Here the n-th term is n × (n + 1).
4. Mixed operation: 5, 11, 23, 47 is "multiply by 2 and add 1".
5. Squares and cubes: 1, 4, 9, 16 or 1, 8, 27, 64, sometimes with 1 added or subtracted.

Number analogy

In an analogy such as 3 : 10 the second number is made from the first by a rule (here square and add 1, since 3² + 1 = 10). Test the rule on every given pair before applying it to the unknown.

Worked examples

1. Mean, median and range. Data: 6, 9, 12, 15, 23. Mean = 65 ÷ 5 = 13. Median = third value = 12. Range = 23 - 6 = 17.
2. Median with an even count. Data: 23, 15, 31, 18, 27, 12, 35, 20. In order: 12, 15, 18, 20, 23, 27, 31, 35. Median = (20 + 23) ÷ 2 = 21.5.
3. Probability. A bag has 5 red, 4 blue and 3 green balls. P(blue) = 4/12 = 1/3, so P(not blue) = 1 - 1/3 = 2/3. Directly, 8 non-blue balls out of 12 = 2/3.
4. Two dice. A sum of 8 arises from (2,6), (3,5), (4,4), (5,3) and (6,2): 5 outcomes out of 36, so the probability is 5/36.
5. Series. 5, 11, 23, 47, 95, ? Each term is double the previous term plus 1, so the next is 95 × 2 + 1 = 191. The differences 6, 12, 24, 48 also double, giving 95 + 96 = 191.

Exam pointers

• Always arrange the data in ascending order before picking the median.
• With an even number of values the median may not be one of the data values.
• Mean, median and mode are usually all placed among the options, so be sure which one is asked.
• For two dice count ordered pairs: (3,5) and (5,3) are different outcomes but (4,4) occurs once.
• Use the complement, 1 - P(E), for "not" and "at least one" questions.
• In a series, confirm the rule with a second method such as the differences before marking the answer.