Statistics

1. Chapter at a glance

• Extension of mean, median and mode from ungrouped to grouped data using frequency distributions.
• Three methods for mean of grouped data: direct method, assumed mean method and step-deviation method; all yield identical results when applied correctly.
• Mode located via modal class (class with maximum frequency) using the given formula; applicable only for unimodal grouped data with equal class sizes.
• Cumulative frequency distributions of “less than” and “more than” type; median found via median class whose cumulative frequency is just greater than or equal to n/2.
• Empirical relation 3 Median = Mode + 2 Mean.
• Choice of method for mean depends on size of xi and fi; step-deviation convenient when di share a common factor.
• Class marks (mid-points) represent observations in each class interval; class size assumed equal for mode and median formulae.
• Ogives (cumulative frequency curves) mentioned but not required for numerical problems in the given text.

2. Definitions, theorems and results

• Mean of grouped data (direct): \(\bar{x}=\frac{\sum f_i x_i}{\sum f_i}\), where \(x_i\) are class marks.
• Assumed mean method: \(\bar{x}=a+\frac{\sum f_i d_i}{\sum f_i}\), \(d_i=x_i-a\).
• Step-deviation method: \(\bar{x}=a+h\frac{\sum f_i u_i}{\sum f_i}\), \(u_i=\frac{x_i-a}{h}\).
• Mode formula (grouped data): \(\text{Mode}=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)h\), where \(l\) = lower limit of modal class, \(h\) = class size, \(f_1\) = frequency of modal class, \(f_0\) = frequency of preceding class, \(f_2\) = frequency of succeeding class.
• Median formula (grouped data): \(\text{Median}=l+\left(\frac{\frac{n}{2}-\text{cf}}{f}\right)h\), where \(l\) = lower limit of median class, cf = cumulative frequency of class preceding median class, \(f\) = frequency of median class, \(h\) = class size.
• Cumulative frequency: frequency obtained by adding frequencies of all preceding classes.
• “Less than” cumulative frequency distribution uses upper class limits; “more than” uses lower class limits.
• Empirical relation: 3 Median = Mode + 2 Mean.
• No theorems/axioms/ lemmas requiring proof are stated in the chapter; only computational formulae and methods are given.

3. Formula sheet

4. Solved-example patterns

• Mean (grouped frequency table given): Identify class marks → choose direct/assumed-mean/step-deviation method according to size of numbers → compute \(\sum f_i x_i\) or \(\sum f_i d_i\) or \(\sum f_i u_i\) → apply chosen formula.
• Mode (grouped table given): Locate modal class (highest \(f\)) → record \(l, h, f_1, f_0, f_2\) → substitute in mode formula.
• Median (grouped table or cumulative frequency given): Convert to “less than” cumulative frequency if needed → locate median class (cf just ≥ n/2) → record \(l, h, f, \text{cf}\) → substitute in median formula.
• Missing frequency problems: Set up equation using given mean/median → solve for unknown frequency/frequencies.
• Comparison/interpretation: Compute two or more measures and state which represents “most frequent” (mode) vs “average” (mean) vs “middle” (median) value.

5. Common mistakes and exam pitfalls

• Using raw data values instead of class marks for \(x_i\).
• Forgetting to convert “less than” cumulative frequencies into proper class intervals before locating median class.
• Taking modal class as the one with highest upper limit instead of highest frequency.
• Applying mode/median formulae when class sizes are unequal (text states formulae assume equal class sizes).
• Sign errors in \(d_i\) or \(u_i\) when assumed mean is chosen; incorrect choice of \(a\) does not affect final mean but calculation errors do.
• Omitting the class preceding/succeeding the modal class when \(f_0\) or \(f_2\) is required.
• Writing cumulative frequency of median class instead of preceding class in the median formula.
• Forgetting that n/2 must be compared with cumulative frequencies, not raw frequencies.
• Units: marks, rupees, cm, hours, etc., must be retained in final answer as per question.