CBSE Class 10 Mathematics Statistics Notes Set 09

Mean

The mean of raw data is the sum of the observations divided by their number. For a frequency table it is \( \bar x=\frac{\sum f_ix_i}{\sum f_i} \), and for grouped data \( x_i \) is the class mark. The assumed-mean (deviation) method uses \( d_i=x_i-a \) and \( \bar x=a+\frac{\sum f_id_i}{\sum f_i} \); the step-deviation method uses \( u_i=\frac{x_i-a}{h} \) and \( \bar x=a+h\frac{\sum f_iu_i}{\sum f_i} \).
If each observation is increased by a constant, or multiplied by one, the mean undergoes the same change. The total of the deviations from the mean is zero. The mean uses every value, so it is pulled by extreme values and cannot be found when an end class is open. When a wrong value is discovered, correct the total (mean x number) and divide again.

Median

Arrange the data in order. For n observations the median is the middle value when n is odd, and the average of the two middle values when n is even. For grouped data, find the class containing the \( \frac N2 \)th observation (median class) from the cumulative frequencies and use \( \text{Median}=l+\frac{\frac N2-cf}{f}\times h \).
The median is not disturbed by extreme values and is the preferred average for wages or wealth. A less-than ogive and a more-than ogive meet at a point whose horizontal coordinate is the median.

Mode and the Empirical Relation

The mode is the value that occurs most often. For grouped data take the class of highest frequency, convert classes to continuous form if needed (subtract 0.5 from each lower limit and add 0.5 to each upper limit for classes such as 5-14, 15-24), and use \( \text{Mode}=l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h \).
For moderately skewed data the three averages are related by Mode = 3 Median - 2 Mean, which is useful for finding a missing average.

Cumulative Frequency Curves

A less-than cumulative table lists the number of observations below each upper class limit; a more-than table lists those at or above each lower limit. Plotting these against the class limits and joining the points gives the two ogives. For an estimate of how many observations lie below a value inside a class, assume uniform spread within the class and interpolate.