- Measures of Central Tendency are statistical methods used to find a single value that represents the entire data set. They are also called averages.
- The main measures are Mean, Median and Mode.
- Mean (Arithmetic Average) = Sum of all values ÷ Number of observations.
It represents the overall average of the data. - Median = The middle value when data is arranged in ascending or descending order.
If the number of observations is even, median is the average of the two middle values. - Mode = The value that occurs most frequently in the data set.
It shows the most common value in the distribution.
Definitions [1]
Define correlation.
Correlation is basically a measure of the relationship between two or more sets of data. It serves a very useful purpose.
Formulae [6]
Formula: Median Under Grouped Data
\[M=\quad l+\frac{i}{f}\left(\frac{N}{2}-c\right)\]
Formula: Mean (Direct Method) Under Grouped Method
\[\begin{array}
{rcl}{\overline{\mathbf{X}}} & {=} & {\sum fx}
\end{array}\]
Formula: Mean (Indirect Method) Under Grouped Method
\[\bar{x}=A\pm\frac{\sum fd}{N}\]
Formula: Mean Under Ungrouped Data(Direct Method)
\[\overline{\mathrm{X}}=\frac{\sum x}{\mathrm{N}}\]
Formula: Mean (Indirect Method) Under Ungrouped Data
\[\overline{\mathrm{X}}=A+\frac{\sum d}{N}\]
Formula: Median Under Ungrouped Data
\[\mathrm{Value~of}\left(\frac{\mathbf{N}+1}{2}\right)\mathrm{th~item}\]
Key Points
Key Points: Comparison of Mean, Median and Mode
- The comparison of mean, median and mode can be understood using a normal distribution curve, which is bell-shaped and symmetrical.
- In a normal distribution, the mean, median and mode are equal and lie at the centre of the distribution.
- Most of the observations are concentrated around the middle value, while very high and very low values are rare.
- The normal curve is symmetrical, meaning half of the values lie above the centre and half lie below it.
- When data is skewed (not symmetrical), the mean, median and mode do not coincide and their values differ.
Key Points: Computing Mean from Ungrouped Data
- Direct Method: Mean is calculated by adding all the values and dividing by the total number of observations.
- In the direct method, raw data is used directly without any changes.
- Indirect Method is used when there are large numbers or big values, to make calculation easier.
- In the indirect method, an assumed mean is selected and each value is reduced by subtracting this constant (coding method).
- The final mean calculated by both direct and indirect methods is the same, only the calculation process is different.
Key Points: Computing Mean from Grouped Data
- In grouped data, individual values are not shown separately. Therefore, the midpoint (class mark) of each class interval is used to calculate the mean.
- Direct Method: Multiply each class midpoint (x) with its frequency (f), find ∑fx, and divide by total frequency (N).
- Indirect Method: An assumed mean (A) is taken from a middle class to simplify calculation. Deviations (d) are calculated from A and multiplied by frequency (f).
- If class intervals are equal, shortcut method using interval width can also be applied.
- Both direct and indirect methods give the same mean, but the indirect method is easier when data values are large or calculations are lengthy.
Key Points: Measures of Central Tendency
Key Points: Median
- Median is a positional average, meaning it depends on the position of values in a data set.
- It is the middle value that divides the data into two equal parts.
- Median is represented by the symbol M.
- For ungrouped data, the values must first be arranged in ascending or descending order.
- If the number of observations is odd, the median is the exact middle value.
- If the number of observations is even, the median is found by taking the average of the two middle values.
- For grouped data, median is found by using the cumulative frequency table, and the class where the middle observation lies is called the median class.
Key Points: Mode
- Mode is the value that occurs most frequently in a data set.
- It is represented by Z or M₀.
- Mode is less commonly used compared to mean and median.
- To find mode in ungrouped data, values are first arranged in ascending or descending order to easily spot repeated values.
- A data set can be unimodal (one mode), bimodal (two modes), trimodal (three modes), multimodal (many modes), or have no mode if no value repeats.
