Statistics — chapter notes

In grouped data each class is represented by its class mark (mid-point):

For equal classes, consecutive class marks differ by the class width $h$. Inclusive classes such as $11$–$20$, $21$–$30$ are made continuous ($10.5$–$20.5$, $20.5$–$30.5$) before using the median or mode formula; their class marks are unchanged ($15.5$, $25.5$).

• Direct: $\bar x = \dfrac{\sum f_ix_i}{\sum f_i}$.
• Assumed mean ($a$ a convenient class mark, $d_i = x_i - a$): $\bar x = a + \dfrac{\sum f_id_i}{\sum f_i}$.
• Step deviation ($u_i = \dfrac{x_i - a}{h}$): $\bar x = a + h \times \dfrac{\sum f_iu_i}{\sum f_i}$.

All three give the same mean; the choice of $a$ does not affect the result. Example: runs data (frequencies $6, 10, 12, 8, 4$ on $0$–$100$ in steps of $20$): $\sum f_ix_i = 1880$, $\sum f_id_i = -120$ ($a = 50$), $\sum f_iu_i = -6$; mean $47$ every time.

Write $\sum f_i$ and $\sum f_ix_i$ (or $\sum f_iu_i$) in terms of the unknown, and use the given mean:

Example: frequencies $5, 8, p, 12, 7$ on $0$–$50$, mean $27$: $\dfrac{880 + 25p}{32 + p} = 27$ gives $p = 8$.

With two unknowns, the total frequency gives a second equation. With the median given, use the median formula, after checking which class contains the median.

The modal class is the class with the highest frequency. With $l$ its lower limit, $f_1$ its frequency, $f_0$ and $f_2$ the frequencies of the classes before and after, and $h$ the class width:

• First class modal: $f_0 = 0$; last class modal: $f_2 = 0$.
• Two classes tied for the highest frequency: bimodal — the formula is not used.
• The mode equals the class mark only when $f_0 = f_2$.

Example: wages $8, 12, 20, 8, 2$ on $200$–$450$: mode $= 300 + \dfrac{8}{20} \times 50 = 320$.

1. Find the cumulative frequencies (cf) and $\dfrac n2$.
2. The median class is the first class whose cf is $\ge \dfrac n2$.
3. With $l$ its lower limit, $cf$ the cumulative frequency of the class before it, $f$ its frequency and $h$ the width:

Example: travel times $8, 14, 20, 12, 6$ on $0$–$50$: cf $8, 22, 42, 54, 60$; median $= 20 + \dfrac{30 - 22}{20} \times 10 = 24$.

Not in the 2026–27 syllabus: the graphical representation of cumulative frequency (ogives). Cumulative frequency tables are used.

• Mean: uses every value; affected by extreme values.
• Median: the middle value; not affected much by a few extremes (use for incomes).
• Mode: the most common value (use for sizes, preferences).

Empirical relation (moderately skewed data): $3\,\text{Median} \approx \text{Mode} + 2\,\text{Mean}$. Example: wages mean $309$, median $312.5$ give mode $\approx 319.5$ (actual $320$).