Mean, median, and mode are different measures of central tendency. But what exactly is the central tendency? It is a summary measure that attempts to describe an entire set of data with a single value, representing the middle or centre of its distribution. Mean, median, and mode are the three key measurements of central tendency.
Surveys, on the other hand, are a common method of collecting data by gathering information from a group of individuals. Surveys are typically conducted through questionnaires or interviews administered to a sample of respondents from a population, and the collected data is then analysed scientifically.
Mean:
The mean is the most commonly used measure of central tendency and represents the average of a given collection of data. It applies to both continuous and discrete data. The mean is calculated by dividing the sum of all values by the total number of values.
Example (Mean for Ungrouped Data): In a class, the Geography marks of five students are 35, 68, 85, 32, and 50. The average mark (mean) is calculated as:
(35 + 68 + 85 + 32 + 50) ÷ 5 = 270 ÷ 5 = 54
Formula: For ungrouped data, the mean (M) is given by:
M=∑XNM = \dfrac{\sum{X}}{N}M=N∑X
Where ∑X\sum{X}∑X is the sum of all values and NNN is the total number of values.
Mean for Grouped Data: For grouped data, the formula is adjusted to include frequencies (f) and midpoints (X) of class intervals:
M=∑fxNM = \dfrac{\sum{fx}}{N}M=N∑fx
Where fff is the frequency, XXX is the midpoint, and NNN is the total frequency.
Example (Mean for Grouped Data):
Score |
Frequency (f) |
Midpoint (X) |
fx |
70-74 |
3 |
72 |
216 |
65-69 |
4 |
67 |
268 |
60-64 |
5 |
62 |
310 |
55-59 |
6 |
57 |
342 |
50-54 |
7 |
52 |
364 |
Total frequency, N=25N = 25N=25, and ∑fx=1500\sum{fx} = 1500∑fx=1500.
Mean = 1500 ÷ 25 = 60
Median:
The median is the value that lies at the midpoint of a data set when the values are arranged in ascending or descending order. It divides the data into two equal parts, where 50% of the values lie below and 50% lie above the median.
Example (Median for Ungrouped Data): Let the scores of nine students in English be: 28, 25, 46, 43, 56, 60, 37, 15, 52.
Arranging the scores in ascending order: 15, 25, 28, 37, 43, 46, 52, 56, 60
The median is the N+12\dfrac{N+1}{2}2N+1-th value, where N=9N = 9N=9, so the median is the 5th value, which is 43.
Median for Grouped Data: For grouped data, the median is calculated using the class interval where N2\dfrac{N}{2}2N-th cumulative frequency lies. The formula is:
Median=L+(N2−Ffm)×i\text{Median} = L + \left( \dfrac{\dfrac{N}{2} – F}{f_m} \right) \times iMedian=L+fm2N−F×i
Where:
- LLL = lower limit of the median class,
- FFF = cumulative frequency before the median class,
- fmf_mfm = frequency of the median class,
- iii = class interval.
Example (Median for Grouped Data):
Score |
Frequency (f) |
Lower limit (L) |
Cumulative frequency (cf) |
70-74 |
3 |
69.5 |
3 |
65-69 |
4 |
64.5 |
7 |
60-64 |
5 |
59.5 |
12 |
55-59 |
6 |
54.5 |
18 |
50-54 |
7 |
49.5 |
25 |
Median = 59.5 + 12.5−75×5=62.20\dfrac{12.5 – 7}{5} \times 5 = 62.20512.5−7×5=62.20
Mode:
Mode is the value that appears most frequently in a data set. It is the point in a distribution that corresponds to the highest frequency.
Example (Mode for Ungrouped Data): For the scores: 30, 35, 25, 40, 35, 45, 40, 35, 28, 34
The mode is 35, as it occurs most frequently.
Formula (Mode for Grouped Data):
Mode=3×Median−2×Mean\text{Mode} = 3 \times \text{Median} – 2 \times \text{Mean}Mode=3×Median−2×Mean
Using previous calculations: Mode = 3 × 62.20 – 2 × 60 = 66.68
Importance of Central Tendency in Surveys:
In surveys, we first collect data through interviews or questionnaires. Central tendency measures (mean, median, and mode) are then used to summarise and analyse this data. They help identify typical values and patterns, giving insights into whether the data is normally distributed or skewed.
Conclusion:
Mean, median, and mode provide valuable insights into the data collected through surveys. The mean and median give a sense of the average values, while the mode highlights the most common value. Survey analysis is incomplete without these measures, as they help interpret the data and guide decision-making.
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