Statistics — Mean, Median, Mode, Standard Deviation

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In competitive exams like SSC CGL, understanding basic statistical concepts is crucial for solving quantitative aptitude problems. This section will cover the fundamental concepts of Mean, Median, Mode, and Standard Deviation, which are essential for data analysis.

1. Mean (Average)

The mean, often referred to as the average, is the sum of all the values in a dataset divided by the number of values in that dataset. It is the most common measure of central tendency.

Calculation of Mean:

For a dataset $x_1, x_2, x_3, \ldots, x_n$, the mean ($\bar{x}$) is calculated as:

$$ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} $$

Where:

• $\sum$ represents the summation (sum) of the values.
• $x_i$ represents each individual value in the dataset.
• $n$ represents the total number of values in the dataset.

Example:

Consider the dataset: 10, 12, 15, 11, 13.

Here, $n = 5$.

Sum of values = $10 + 12 + 15 + 11 + 13 = 61$.

Mean ($\bar{x}$) = $\frac{61}{5} = 12.2$.

Mean for Grouped Data:

For grouped data, where values are presented in a frequency distribution, the mean is calculated by summing the product of each value (or midpoint of a class interval) and its corresponding frequency, then dividing by the total frequency.

For a dataset with values $x_1, x_2, \ldots, x_k$ and their corresponding frequencies $f_1, f_2, \ldots, f_k$:

$$ \bar{x} = \frac{\sum_{i=1}^{k} f_i x_i}{\sum_{i=1}^{k} f_i} = \frac{f_1 x_1 + f_2 x_2 + \ldots + f_k x_k}{f_1 + f_2 + \ldots + f_k} $$

Example for Grouped Data:

Consider the following data:

Value ($x_i$)	Frequency ($f_i$)
5	2
10	3
15	5
20	1

Total frequency ($\sum f_i$) = $2 + 3 + 5 + 1 = 11$.

Sum of ($f_i x_i$) = $(2 \times 5) + (3 \times 10) + (5 \times 15) + (1 \times 20) = 10 + 30 + 75 + 20 = 135$.

Mean ($\bar{x}$) = $\frac{135}{11} \approx 12.27$.

2. Median

The median is the middle value in a dataset that has been ordered from least to greatest. It divides the dataset into two equal halves. The median is less affected by outliers (extremely high or low values) than the mean.

Calculation of Median:

The method of calculating the median depends on whether the number of observations ($n$) is odd or even.

1. If $n$ is odd: The median is the value of the $\left(\frac{n+1}{2}\right)^{th}$ observation.
2. If $n$ is even: The median is the average of the values of the $\left(\frac{n}{2}\right)^{th}$ and $\left(\frac{n}{2} + 1\right)^{th}$ observations.

Example (Odd $n$):

Consider the dataset: 3, 1, 4, 2, 5.

First, order the data: 1, 2, 3, 4, 5.

Here, $n = 5$ (odd).

The median is the $\left(\frac{5+1}{2}\right)^{th} = 3^{rd}$ observation.

Median = 3.

Example (Even $n$):

Consider the dataset: 6, 2, 8, 4, 10, 12.

First, order the data: 2, 4, 6, 8, 10, 12.

Here, $n = 6$ (even).

The median is the average of the $\left(\frac{6}{2}\right)^{th} = 3^{rd}$ and $\left(\frac{6}{2} + 1\right)^{th} = 4^{th}$ observations.

The 3rd observation is 6, and the 4th observation is 8.

Median = $\frac{6 + 8}{2} = \frac{14}{2} = 7$.

Median for Grouped Data:

For grouped data, the median is found by first identifying the median class. The median class is the class interval whose cumulative frequency is just greater than or equal to $\frac{n}{2}$, where $n$ is the total frequency.

The formula for the median of grouped data is:

$$ \text{Median} = L + \left(\frac{\frac{n}{2} - CF}{f}\right) \times h $$

Where:

• $L$ is the lower limit of the median class.
• $n$ is the total frequency ($\sum f_i$).
• $CF$ is the cumulative frequency of the class preceding the median class.
• $f$ is the frequency of the median class.
• $h$ is the class width (size) of the median class.

Example for Grouped Data:

Consider the following frequency distribution:

Class Interval	Frequency ($f$)	Cumulative Frequency ($CF$)
0-10	5	5
10-20	10	15
20-30	25	40
30-40	30	70
40-50	15	85

Total frequency ($n$) = 85.

We need to find the class interval where the cumulative frequency is just greater than or equal to $\frac{n}{2} = \frac{85}{2} = 42.5$.

The cumulative frequency 40 is less than 42.5, but the next cumulative frequency, 70, is greater than 42.5. Therefore, the median class is 30-40.

Now, apply the formula:

• $L = 30$ (lower limit of the median class)
• $n = 85$
• $CF = 40$ (cumulative frequency of the class preceding the median class, which is 20-30)
• $f = 30$ (frequency of the median class, which is 30-40)
• $h = 10$ (class width, 40 - 30)

Median = $30 + \left(\frac{42.5 - 40}{30}\right) \times 10 = 30 + \left(\frac{2.5}{30}\right) \times 10 = 30 + \frac{25}{30} = 30 + \frac{5}{6} \approx 30 + 0.833 = 30.833$.

3. Mode

The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), more than one mode (multimodal), or no mode at all.

Calculation of Mode:

• Ungrouped Data: The mode is simply the value that occurs most often.
• Grouped Data: The mode is calculated using the modal class, which is the class interval with the highest frequency.

Example (Ungrouped Data):

Dataset: 2, 3, 3, 4, 5, 5, 5, 6, 7.

The value 5 appears three times, which is more than any other value. So, the mode is 5.

Dataset: 1, 2, 2, 3, 4, 4, 5.

This dataset is bimodal, with modes 2 and 4.

Dataset: 1, 2, 3, 4, 5.

Each value appears only once, so there is no mode.

Mode for Grouped Data:

The formula for the mode of grouped data is:

$$ \text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h $$

Where:

• $L$ is the lower limit of the modal class.
• $f_1$ is the frequency of the modal class.
• $f_0$ is the frequency of the class preceding the modal class.
• $f_2$ is the frequency of the class succeeding the modal class.
• $h$ is the class width of the modal class.

Example for Grouped Data:

Consider the same frequency distribution as used for the median:

Class Interval	Frequency ($f$)
0-10	5
10-20	10
20-30	25
30-40	30
40-50	15

The modal class is the class with the highest frequency, which is 30-40 (frequency = 30).

Now, apply the formula:

• $L = 30$ (lower limit of the modal class)
• $f_1 = 30$ (frequency of the modal class)
• $f_0 = 25$ (frequency of the class preceding the modal class, which is 20-30)
• $f_2 = 15$ (frequency of the class succeeding the modal class, which is 40-50)
• $h = 10$ (class width)

Mode = $30 + \left(\frac{30 - 25}{2 \times 30 - 25 - 15}\right) \times 10 = 30 + \left(\frac{5}{60 - 40}\right) \times 10 = 30 + \left(\frac{5}{20}\right) \times 10 = 30 + \frac{50}{20} = 30 + 2.5 = 32.5$.

4. Standard Deviation (SD)

Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

Calculation of Standard Deviation:

The standard deviation is the square root of the variance. Variance is the average of the squared differences from the Mean.

For Ungrouped Data:

Let $\bar{x}$ be the mean of the dataset $x_1, x_2, \ldots, x_n$. The standard deviation ($\sigma$) is:

$$ \sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}} $$

The term $\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$ is the variance ($\sigma^2$).

Example (Ungrouped Data):

Dataset: 2, 4, 4, 4, 5, 5, 7, 9.

First, calculate the mean:

Sum = $2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40$.

Number of values ($n$) = 8.

Mean ($\bar{x}$) = $\frac{40}{8} = 5$.

Now, calculate the squared differences from the mean:

• $(2-5)^2 = (-3)^2 = 9$
• $(4-5)^2 = (-1)^2 = 1$
• $(4-5)^2 = (-1)^2 = 1$
• $(4-5)^2 = (-1)^2 = 1$
• $(5-5)^2 = (0)^2 = 0$
• $(5-5)^2 = (0)^2 = 0$
• $(7-5)^2 = (2)^2 = 4$
• $(9-5)^2 = (4)^2 = 16$

Sum of squared differences = $9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32$.

Variance ($\sigma^2$) = $\frac{32}{8} = 4$.

Standard Deviation ($\sigma$) = $\sqrt{4} = 2$.

For Grouped Data:

For grouped data with frequencies $f_1, f_2, \ldots, f_k$ for values $x_1, x_2, \ldots, x_k$ and mean $\bar{x}$:

$$ \sigma = \sqrt{\frac{\sum_{i=1}^{k} f_i (x_i - \bar{x})^2}{\sum_{i=1}^{k} f_i}} $$

Alternatively, a more convenient formula for calculation is:

$$ \sigma = \sqrt{\frac{\sum f_i x_i^2}{\sum f_i} - \left(\frac{\sum f_i x_i}{\sum f_i}\right)^2} = \sqrt{\frac{\sum f_i x_i^2}{N} - (\bar{x})^2} $$

Where $N = \sum f_i$ is the total frequency.

Example for Grouped Data:

Consider the following data:

Value ($x_i$)	Frequency ($f_i$)	$f_i x_i$	$x_i^2$	$f_i x_i^2$
2	3	6	4	12
4	5	20	16	80
6	2	12	36	72

Total frequency ($N$) = $3 + 5 + 2 = 10$.

Sum of ($f_i x_i$) = $6 + 20 + 12 = 38$.

Sum of ($f_i x_i^2$) = $12 + 80 + 72 = 164$.

Mean ($\bar{x}$) = $\frac{\sum f_i x_i}{N} = \frac{38}{10} = 3.8$.

Now, use the formula for standard deviation:

$$ \sigma = \sqrt{\frac{\sum f_i x_i^2}{N} - (\bar{x})^2} = \sqrt{\frac{164}{10} - (3.8)^2} $$

$$ \sigma = \sqrt{16.4 - 14.44} = \sqrt{1.96} = 1.4 $$

So, the standard deviation is 1.4.

Key Takeaways for Exams:

• Mean: Sensitive to outliers. Useful for symmetrical data.
• Median: Not sensitive to outliers. Useful for skewed data.
• Mode: Represents the most common value. Useful for categorical data or identifying peaks.
• Standard Deviation: Measures spread. A higher SD means more variability.
• Relationship: Remember the empirical relationship for skewed distributions.

Practice solving problems from previous years' papers to get comfortable with the application of these concepts under timed conditions.