Central Tendency - Practice Questions & MCQ

Central Value of Data

A measure of central tendency (or central value) is a single value that attempts to describe a set of data by identifying the central position within that set of data. Apart from the mean (often called the average), there are other central values such as the median and the mode.

The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. 

Mean

The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. If we have $n$ values in a data set, i.e. $x_1, x_2, x_3, \ldots, x_n$, then its mean, usually denoted by $\bar{x}$ (pronounced "x bar"), is:

$
\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}
$

Median

The median is the middle value for a set of data that has been arranged in ascending or descending order. 

For example, to find the median of the following data

$
\begin{array}{lllllllllll}
65 & 55 & 89 & 56 & 35 & 14 & 56 & 55 & 87 & 45 & 92
\end{array}
$

We first rearrange that data into order (ascending)

$
\begin{array}{lllllllllll}
14 & 35 & 45 & 55 & 55 & 56 & 56 & 65 & 87 & 89 & 92
\end{array}
$

The median mark is the value exactly in the middle - in this case, 56

When the n is even in the data set, then simply you have to take the middle two scores and average them.

Mode

The mode is the most frequent value in our data set.

Normally, the mode is used for categorical data where we wish to know which is the most common category, 

$
\begin{array}{llllllllllll}
65 & 55 & 89 & 56 & 35 & 14 & 56 & 55 & 87 & 45 & 92 & 55
\end{array}
$

In the above case, the mode of the data set is 55.

MEAN (Arithmetic Mean)

The mean is the sum of the value of each observation in a dataset divided by the number of observations.

For example, to calculate the mean weight of 50 people, add the 50 weights together and divide by 50. Technically this is the arithmetic mean. 

Mean of the Ungrouped Data 

$
\begin{aligned}
&\text { If } \mathrm{n} \text { observations in data are } \mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \ldots \ldots, \mathrm{x}_{\mathrm{n}} \text {, then arithmetic mean } \bar{x} \text { is given by }\\
&\bar{x}=\frac{x_1+x_2+x_3+\ldots \cdots+x_n}{n}=\frac{1}{n} \sum_{i=1}^n x_i
\end{aligned}
$

Mean of Ungrouped Frequency Distribution

If observations in data are $x_1, x_2, x_3, \ldots \ldots, x_n$ with respective frequencies $f_1, f_2, f_3, \ldots \ldots, f_n$; then
Some of the value of the observations $=f_1 x_1+f_2 x_2+f_3 x_3+\ldots \ldots .+f_n x_n$
and Number of observations $=f_1+f_2+f_3+\ldots \ldots+f_n$
The mean in this case is given by

$
\bar{x}=\frac{f_1 x_1+f_2 x_2+f_3 x_3+\ldots \ldots+f_n x_n}{f_1+f_2+f_3+\ldots \ldots+f_n}=\frac{\sum_{i=1}^n f_i x_i}{\sum_{i=1}^n f_i}
$

Grouped Frequency Distribution

$x_i$ is taken as the mid-point of respective classes (or intervals).
i.e.,

$
m=\frac{\text { lower boundary }+ \text { upper boundary }}{2}
$

then, $\bar{x}=\frac{\sum_{i=1}^n f_i m_i}{\sum_{i=1}^n f_i}$

For example,

A frequency table displaying the professor’s last statistic test is shown, the best estimate of the class mean is 

$
\begin{array}{|c|c|}
\hline \text { Grade Interval } & \text { Number of Students } \\
\hline 10-12 & 1 \\
\hline 12-14 & 2 \\
\hline 14-16 & 0 \\
\hline 16-18 & 4 \\
\hline 18-20 & 1 \\
\hline
\end{array}
$

First, find the midpoints for all intervals.

$
\begin{array}{|c|c|}
\hline \text { Grade Interval } & \text { Midpoint } \\
\hline 10-12 & 11 \\
\hline 12-14 & 13 \\
\hline 14-16 & 15 \\
\hline 16-18 & 17 \\
\hline 18-20 & 19 \\
\hline
\end{array}
$

$
\begin{aligned}
&\text { Now calculate the sum of the product of each interval frequency and midpoint, } \sum_{i=i}^n f_i m_i\\
&\begin{aligned}
& 11(1)+13(2)+15(0)+17(4)+19(1)=124 \\
& \bar{x}=\frac{\sum_{i=1}^n f_i m_i}{\sum_{i=1}^n f_i}=\frac{124}{8}=15.5
\end{aligned}
\end{aligned}
$

MEDIAN

The median is the "middle value" of the data when arranged in ascending or descending order.

It is a number that separates ordered data into 2 equal halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger.

Median of Ungrouped Data

If the number of observations is n,

First, arrange the observations in ascending or descending order.

If n is odd :       

$
\text { Median }=\left(\frac{n+1}{2}\right)^{t h} \text { observation }
$

If n is even :             

$
\text { Median }=\frac{\text { Value of }\left(\frac{n}{2}\right)^{\text {th }} \text { observation }+ \text { Value of }\left(\frac{n}{2}+1\right)^{t h} \text { observation }}{2}
$

For example,

Consider the following data: $1 ; 11.5 ; 6 ; 7.2 ; 4 ; 8 ; 9 ; 10 ; 6.8 ; 8.3 ; 2 ; 2 ; 10 ; 1$
Ordered from smallest to largest: : $1 ; 1 ; 2 ; 2 ; 4 ; 6 ; 6.8 ; 7.2 ; 8 ; 8.3 ; 9 ; 10 ; 10 ; 11.5$
Since there are 14 observations, the median is an average of $(n / 2)$ th $=7$ th and $(n / 2+1)$ th $=8$ th term. So the median is the average of 6.8 and 7.2, which equals 7.
The median is seven. Half of the values are smaller than seven and half of the values are larger than seven.
Median of Ungrouped Frequency Distribution
To find the median, first arrange the observations in ascending order. After this, the cumulative frequencies are obtained.
Let the sum of frequencies be denoted by N.

$
\underline{N+1}
$
Now if N is odd, then identify the observation whose cumulative frequency is equal to or just greater than $\ \ \frac {N+1}{2}$. This value of the observation lies in the middle of the data and therefore, it is the required median.

Median of Continuous Frequency Distribution

In this case, the following formula can be used.

1. When observations arranged in ascending order   

Median $=l+\frac{\left(\frac{N}{2}-c f\right)}{f} \times h$
where,
I = lower limit of median class,
$\mathrm{N}=$ number of observations,
$\mathrm{cf}=$ cumulative frequency of class preceding the median class,
$\mathrm{f}=$ frequency of median class,
$\mathrm{h}=$ class size (width) (assuming class size to be equal).

MODE

Mode is that value among the observations which occurs most often, that is, the value of the observation having the maximum frequency. 

In a grouped frequency distribution, it is not possible to determine the mode by looking at the frequencies. Here, we can only locate a class with the maximum frequency, called the modal class. The mode is a value inside the modal class, and is given by the formula:

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

where 

l = lower limit of the modal class, 

h = size of the class interval (assuming all class sizes to be equal), 

f₁ = frequency of the modal class, 

f₀ = frequency of the class preceding the modal class, 

f₂ = frequency of the class succeeding the modal class.