0JEE Mains statistics, understanding mean, median, and variance is crucial. The mean is the average of a dataset, calculated by summing all values and dividing by the number of values. The median is the middle value when the data is arranged in order. Variance measures how spread out the data is from the mean, calculated by averaging the squared differences between each data point and the mean.
Mean:
- Ungrouped data: Sum of all values divided by the number of values.
- Grouped data: Sum of (midpoint of each class * frequency) divided by the total frequency.
Median:
- Ungrouped data:The middle value when data is arranged in ascending order. If there are two middle values, their average is the median.
- Grouped data:Calculated using the formula: l + ((N/2 - CF) / f) * h, where:
- l is the lower limit of the median class.
- N is the total number of observations.
- 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.
- l is the lower limit of the median class.
Variance:
- Ungrouped data: Average of the squared differences between each data point and the mean.
- Grouped data: Average of the squared differences between the midpoint of each class and the mean, weighted by their respective frequencies.
Important points:
- Variance is always calculated about the mean.
- Standard deviation is the square root of the variance.
- These measures are used to understand the spread and distribution of data.
