Tales by Dots and Lines – Class 8 Mathematics notes, important questions and MCQs

Chapter overview

The mean and the median summarise data in a single number. The mean can be seen as the balance point of the data, and it changes in predictable ways when values change. The median is the middle value. Both can be found from frequency tables and used to find unknown values.

The mean as a balance point

• Mean = sum of values ÷ number of values; it is the point where the data 'balances'.
• The sum of the differences of all values from the mean is zero (values above balance values below).
• Adding the same number k to every value increases the mean by k; multiplying every value by k multiplies the mean by k.
• If the mean of n values is m, their sum is n × m; this helps find a missing value.

The median

• Arrange data in order; the median is the middle value (or the average of the two middle values for an even count).
• Changing the largest value does not change the median, but it changes the mean.
• From a frequency table, the mean is Σ(f × x) ÷ Σf; the median is found by locating the middle position using cumulative frequencies.

Key terms

Mean

the sum of the values divided by the number of values; the balance point of the data

Median

the middle value of data arranged in order

Frequency

the number of times a value occurs in the data

Deviation

the difference between a value and the mean

Formulas

• Mean = Σx ÷ n
• Mean from a frequency table = Σ(f × x) ÷ Σf
• Sum of values = mean × number of values
• Σ(x − mean) = 0

Important questions with answers

Quick revision

• Mean = balance point; deviations add to zero.
• Add k to all → mean + k; multiply all by k → mean × k.
• Median = middle value after ordering.
• Frequency mean = Σfx ÷ Σf.

Exam tips

• To find a missing value, first find the total from the mean, then subtract the known values.
• When a question changes one extreme value, ask: does the middle value change? Usually the median stays the same.

Practice MCQs

1. The mean of 2, 4, 6, 8 and 10 is

   1. 5
   2. 6
   3. 7
   4. 30
   Show answer

   Answer: (b) 6. Sum 30 ÷ 5 = 6.

2. The median of 1, 3, 4, 8, 9 is

   1. 3
   2. 4
   3. 5
   4. 8
   Show answer

   Answer: (b) 4. The middle (3rd) value is 4.

3. The number of times a value occurs in data is its

   1. mean
   2. median
   3. frequency
   4. range
   Show answer

   Answer: (c) frequency. Frequency counts occurrences.

4. The mean of 5 numbers is 12. Their sum is

   1. 17
   2. 2.4
   3. 60
   4. 12
   Show answer

   Answer: (c) 60. 5 × 12 = 60.

5. If 3 is added to each of the values 4, 6, 8, the new mean is

   1. 6
   2. 3
   3. 18
   4. 9
   Show answer

   Answer: (d) 9. The old mean 6 increases by 3.

6. The mean of 4, 7, x and 9 is 7. The value of x is

   1. 6
   2. 7
   3. 8
   4. 28
   Show answer

   Answer: (c) 8. Sum = 28, so x = 28 − 20 = 8.

7. Values 2, 4, 6 have mean 4. The sum of their deviations from the mean is

   1. 4
   2. 2
   3. 12
   4. 0
   Show answer

   Answer: (d) 0. (−2) + 0 + 2 = 0.

8. In a table, value 1 occurs 2 times, value 2 occurs 3 times and value 3 occurs 5 times. The mean is

   1. 2
   2. 2.3
   3. 2.5
   4. 3
   Show answer

   Answer: (b) 2.3. Σfx = 2 + 6 + 15 = 23, Σf = 10, mean = 2.3.