Chapter 13: Statistics

In Class IX you computed the mean, median, and mode of small ungrouped data sets. In real life, data sets are often huge , height of every student in a school, marks of every candidate in an exam , and listing every value is impractical. Statisticians group the data into intervals (or classes) and report the frequency of each interval. Now the three measures of central tendency must be computed from the grouped data, using slightly more involved formulas.

This chapter introduces those formulas and the procedures to apply them. You will learn three methods to find the mean (direct, assumed mean, step-deviation), the formula for the median of grouped data, and the formula for the mode. You will also draw cumulative frequency curves, called ogives, which let you read off the median graphically.

For the board exam, this is a comfortable chapter , the calculations are mechanical, the formulas are short, and once you tabulate the data correctly the answers fall out. The most common mistakes are arithmetic ones (sum a column wrong, mismatch a row), so neatness pays off here more than in any other chapter.

In real life, statistics is the foundation of every data-driven decision: clinical trials, public health, weather forecasts, economic policy, business analytics, sports analytics. The simple measures here , mean, median, mode , are the starting point.

What's inside

• Grouped data and class marks , the language of frequency tables.
• Mean of grouped data , direct, assumed mean, and step-deviation methods.
• Mode of grouped data , the modal class and its formula.
• Median of grouped data , the median class and the cumulative frequency.
• Ogive , cumulative frequency curve and median from a graph.

Key results / Formula card

For a frequency table with class marks $x_i$ and frequencies $f_i$:

• Mean (direct method): $\bar x = \dfrac{\sum f_i x_i}{\sum f_i}$.
• Mean (assumed mean): pick a guess $a$, compute $d_i = x_i - a$, then $\bar x = a + \dfrac{\sum f_i d_i}{\sum f_i}$.
• Mean (step-deviation): with class size $h$, compute $u_i = (x_i - a)/h$, then $\bar x = a + h \cdot \dfrac{\sum f_i u_i}{\sum f_i}$.
• Mode: $\text{Mode} = L + \dfrac{f_1 - f_0}{2 f_1 - f_0 - f_2} \times h$, where $L$ is the lower limit of the modal class, $f_1$ the modal frequency, $f_0$ and $f_2$ frequencies of preceding and succeeding classes, $h$ the class size.
• Median: $\text{Median} = L + \dfrac{(n/2) - F}{f} \times h$, where $L$ is the lower limit of the median class, $n$ is total frequency, $F$ is the cumulative frequency before the median class, $f$ is the median class's frequency, $h$ is the class size.
• Empirical relation: approximately, $\text{Mode} \approx 3 \cdot \text{Median} - 2 \cdot \text{Mean}$.