Statistics — Mathslive.ie

Section 1 of 4

Words & Data Types

Data is information. Primary data you collect yourself (a survey); secondary data comes from others (the internet).

The two families

Numerical (numbers): discrete = counts (whole, e.g. number of cars); continuous = measured over a range (e.g. weight). Categorical (labels): ordinal = has an order (grades); nominal = no order (colour).

Worked example — classify the data

Name the data type for each: (i) exam grades A, B, C; (ii) car colour; (iii) a person’s weight; (iv) number of siblings.

(i) grades have an order → ordinal categorical

(ii) colour has no order → nominal categorical

(iii) weight is measured over a range → continuous numerical

(iv) siblings are whole counts → discrete numerical

ordinal, nominal, continuous, discrete

Section 2 of 4

The Three Averages

Mean, median, mode

Mean $=\dfrac{\text{sum}}{\text{how many}}$. Median = the middle value (put them in order first). Mode = the most common value.

Worked example — mean, median and mode of a list

Find the mean, median and mode of $3, 5, 2, 9, 4, 6$.

Order: $2,3,4,5,6,9$

Mean $=\dfrac{2+3+4+5+6+9}{6} = \dfrac{29}{6} \approx 4.8$

Median = middle of six values = $\dfrac{4+5}{2} = 4.5$

Mode = most common = none (all appear once)

mean $\approx 4.8$, median $= 4.5$, no mode

Section 3 of 4

Grouped Data

With grouped data, use the mid-interval value of each class as its score.

The marks of $25$ candidates are grouped like this:

Marks	0–40	40–60	60–80	80–100
No. of students	3	9	9	4

Worked example — the mid-interval mean

Taking mid-interval values, calculate the mean mark.

Mid-intervals: $20, 50, 70, 90$

Mean $=\dfrac{(20)(3)+(50)(9)+(70)(9)+(90)(4)}{3+9+9+4}$

$=\dfrac{60+450+630+360}{25} = \dfrac{1500}{25}$

Mean $= 60$

Now you try

You try

For the same $25$ candidates, which class contains the median?

Pen and paper out — try it before you reveal.

Median position $=\dfrac{25+1}{2} = 13\text{th}$ value

$3$ are in $0\text{-}40$, then $9$ more reaches the $12\text{th}$; the $13\text{th}$ is in the next class

the $60\text{-}80$ class

Section 4 of 4

Measures of Spread

Range & interquartile range

Range = highest $-$ lowest. IQR = $Q_3 - Q_1$, where $Q_1$ is at position $\dfrac{n+1}{4}$ and $Q_3$ at $\dfrac{3(n+1)}{4}$.

You try

Find the range and interquartile range of $2, 3, 4, 5, 6, 9$.

Pen and paper out — try it before you reveal.

Range $= 9 - 2 = 7$

$Q_1$ at $\dfrac{6+1}{4} = 1.75 \Rightarrow$ between the $1$st ($2$) and $2$nd ($3$): $Q_1 = 2.5$

$Q_3$ at $\dfrac{3(7)}{4} = 5.25 \Rightarrow$ between the $5$th ($6$) and $6$th ($9$): $Q_3 = 7.5$

IQR $= 7.5 - 2.5$

range $= 7$, IQR $= 5$

You try

Find the interquartile range of $3, 5, 7, 8, 11$.

Pen and paper out — try it before you reveal.

$Q_1$ at $\dfrac{5+1}{4} = 1.5 \Rightarrow$ between $3$ and $5$: $Q_1 = 4$

$Q_3$ at $\dfrac{3(6)}{4} = 4.5 \Rightarrow$ between $8$ and $11$: $Q_3 = 9.5$

IQR $= 9.5 - 4$

IQR $= 5.5$

More examples

Worked example — all three averages

Find the mode, median and mean of $2, 3, 5, 7, 2$.

Mode = most common = $2$

Order $2,2,3,5,7$; middle (the $3$rd) $= 3$

Mean $= \dfrac{2+3+5+7+2}{5} = \dfrac{19}{5}$

mode $=2$, median $=3$, mean $=3.8$

You try

Name the data type: (i) family size; (ii) age; (iii) exam grades; (iv) gender.

Numerical (discrete/continuous) or categorical (ordinal/nominal)?

(i) whole counts → discrete numerical

(ii) measured → continuous numerical

(iii) ordered → ordinal categorical

(iv) no order → nominal categorical

discrete, continuous, ordinal, nominal

That’s Statistics.

Data types, the three averages, grouped-data means, and spread (range & IQR) — the core of Junior Cert statistics.