Mean and Median
ICSE Class 9 · Mathematics · Chapter 18
What this chapter covers
- Averages and central tendency
- The arithmetic mean of raw data
- Using the mean: missing, added, removed and corrected observations
- Properties of the arithmetic mean
- The mean of an ungrouped frequency distribution
- The median of raw data
- The median of an ungrouped frequency distribution
- Choosing between the mean and the median
Key points
- A measure of central tendency is one value that represents a whole set of data; the mean and the median are the two studied in Class IX.
- Ungrouped data keeps every observation at its exact value; it may be raw, arrayed or shown in an ungrouped frequency table.
- Mean of raw data: barx = (Sigma x)/(n), so the total of the observations is Sigma x = nbarx.
- For missing, added, removed or wrongly copied observations, work with totals, never with the means directly.
- The mean of two groups of different sizes is the combined total divided by the combined number, not the average of the two means.
- The sum of the deviations of the observations from their mean is zero.
- Adding a to every observation adds a to the mean; multiplying every observation by k multiplies the mean by k.
- Mean of an ungrouped frequency distribution: barx = (Sigma fx)/(Sigma f); divide by the total frequency, not by the number of rows.
- The median is the middle value of arrayed data: the ((n + 1)/(2))th observation when n is odd, the mean of the ((n)/(2))th and ((n)/(2) + 1)th observations when n is even.
- Always arrange the data before finding the median.
- In a frequency table, use cumulative frequencies to locate the middle position; the first c.f. equal to or greater than the position gives the median's row.
- The mean uses every value and is pulled by extreme values; the median depends only on position and suits data with a few very large or very small values.
Questions answered
Define the arithmetic mean of a set of observations and write its formula for raw data.
The arithmetic mean is the sum of all the observations divided by the number of observations; barx = (Sigma x)/(n).
Define the median of a set of observations.
The median is the value of the middle observation when the observations are arranged in ascending or descending order; for an even number of observations it is the mean of the two middle values.
Find the mean of 7, 11, 14, 18 and 20.
Sigma x = 70 and n = 5, so barx = 14.
Find the median of 13, 7, 22, 9, 16 and 11.
Arranged: 7, 9, 11, 13, 16, 22. Here n = 6, so the median is (11 + 13)/(2) = 12.
Find the mean of the first eight even natural numbers.
The numbers are 2, 4, 6, …, 16 with sum 72, so the mean is 72 ÷ 8 = 9.
The mean of 12 observations is 15. It is later found that the observation 37 was copied as 13. Find the correct mean.
Correct total = 12 × 15 - 13 + 37 = 204, so the correct mean is 204 ÷ 12 = 17.
The mean of five numbers is 20. When one number is excluded, the mean of the remaining four is 18. Find the excluded number.
5 × 20 - 4 × 18 = 100 - 72 = 28.
The mean of 10 observations is 15. Each observation is decreased by 3. Find the new mean, and give the property you used.
The new mean is 15 - 3 = 12, because subtracting a constant from every observation subtracts the same constant from the mean.
Past board exam questions
ICSE 2019 sample paper · 4 marks
Find: (i) Mean and (ii) Median for the following observations: 10, 47, 3, 9, 17, 27, 4, 48, 12, 15
Show answer
(i) Sum = 10 + 47 + 3 + 9 + 17 + 27 + 4 + 48 + 12 + 15 = 192; n = 10.
Mean = (192)/(10) = 19.2
(ii) In ascending order: 3, 4, 9, 10, 12, 15, 17, 27, 47, 48.
n = 10 (even), so median = mean of the 5th and 6th observations = (12 + 15)/(2) = 13.5