Statistics Mean

$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$

The direct method formula for mean is $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$, where $f_i$ are frequencies and $x_i$ are class marks.

Average of upper and lower limits

Class mark $x_i = \frac{\text{lower limit} + \text{upper limit}}{2}$.

Direct, assumed mean, or step-deviation methods

All three methods—direct, assumed mean, and step-deviation—can be used to find the mean, and they give the same result.

Class marks and frequencies

In grouped data, we use class marks ($x_i$) and their corresponding frequencies ($f_i$) to compute the mean.

False

The mean is not necessarily an observation in the data set; it is a calculated average and may not coincide with any data point.

$\frac{x_i - a}{h}$

In step-deviation method, $u_i = \frac{x_i - a}{h}$, where $a$ is assumed mean and $h$ is class size.

500

Using $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$, we get $\sum f_i x_i = \bar{x} \times \sum f_i = 25 \times 20 = 500$.

Step-deviation method

Step-deviation method simplifies calculations when class marks are large by using deviations from an assumed mean.

$x_i - a$

In assumed mean method, $d_i = x_i - a$, where $a$ is the assumed mean.

Make the classes continuous by adjusting the limits

For grouped data with discontinuous classes, we first make the classes continuous by subtracting and adding half the gap to lower and upper limits respectively.

True

The step-deviation method is most convenient when class sizes are equal, as it simplifies calculations using a common factor $h$.

Zero

A key property of mean is that the sum of deviations of all observations from the mean is zero: $\sum (x_i - \bar{x}) = 0$.

Increases by 5

If each observation is increased by a constant $k$, the mean also increases by $k$. So the new mean is original mean + 5.

18.89

Sum of 10 observations = $10 \times 20 = 200$. After removing 30, sum = 170. New mean = $\frac{170}{9} \approx 18.89$.