Statistics Class 10: notes and important questions

Mean of grouped data

Use the class mark \(x_i=\dfrac{\text{lower limit}+\text{upper limit}}{2}\) as the representative value of each class.

• Direct method: \(\bar x=\dfrac{\sum f_ix_i}{\sum f_i}\)
• Assumed mean method: choose \(a\) (a central \(x_i\)), let \(d_i=x_i-a\); \(\bar x=a+\dfrac{\sum f_id_i}{\sum f_i}\)
• Step-deviation method (equal class width \(h\)): \(u_i=\dfrac{x_i-a}{h}\); \(\bar x=a+h\cdot\dfrac{\sum f_iu_i}{\sum f_i}\)

All three give the same answer. The last two just reduce the arithmetic.

Mode of grouped data

The modal class is the class with the highest frequency. Then

\(\text{Mode}=l+\left(\dfrac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h\)

where \(l\) is the lower limit of the modal class, \(f_1\) its frequency, \(f_0\) and \(f_2\) the frequencies of the classes before and after it, and \(h\) the class width.

Median of grouped data

Find the cumulative frequencies (cf) and \(\frac n2\), where \(n=\sum f_i\). The median class is the first class whose cf is greater than (or equal to) \(\frac n2\). Then

\(\text{Median}=l+\left(\dfrac{\frac n2-cf}{f}\right)\times h\)

where \(l\) is the lower limit of the median class, \(cf\) the cumulative frequency of the class before it, \(f\) its frequency, and \(h\) the class width.

Empirical relationship

\(3\,\text{Median}=\text{Mode}+2\,\text{Mean}\) (approximately true for moderately skewed data).

Worked example 1 (step deviation)

Classes 10–20, 20–30, 30–40, 40–50 with \(f=3,5,8,4\). With \(a=35\) and \(h=10\): \(u=-2,-1,0,1\) and \(\sum fu=-6-5+0+4=-7\), \(n=20\). Mean \(=35+10\left(\frac{-7}{20}\right)=31.5\).

Worked example 2 (mode)

For the same data, the modal class is 30–40: \(l=30,\ f_1=8,\ f_0=5,\ f_2=4\). Mode \(=30+\dfrac{3}{16-9}\times10=30+\dfrac{30}{7}\approx34.29\).

Worked example 3 (median)

cf: 3, 8, 16, 20; \(\frac n2=10\), so the median class is 30–40 with \(cf=8\), \(f=8\). Median \(=30+\dfrac{10-8}{8}\times10=32.5\).

Common errors

• Using the cf of the median class itself instead of the class before it.
• Taking \(f_0,f_2\) from the wrong classes, or using the class with the largest cf as the modal class.
• Applying the formulas to non-continuous classes (1–10, 11–20, …). First convert them to 0.5–10.5, 10.5–20.5, …
• Forgetting to multiply by \(h\) in the step-deviation method.

Board-exam tips

• Always show a full table with columns \(x_i\), \(f_ix_i\) (or \(u_i\), \(f_iu_i\)) and cf. The table carries method marks.
• For missing frequencies, set up two equations: one from \(\sum f_i=n\) and one from the given mean or median.
• Bimodal data is not asked. Graphical representation (ogives) is deleted from the current syllabus.