Presenting Data — Mathslive.ie

Section 1 of 6

Histogram

A histogram shows grouped (continuous) data as bars. Bar height = frequency.

Bars touch — intervals are continuous, not separate categories.

(i) Worked example

Ages of 20 people:

Age	0–2	2–4	4–6	6–8
People	5	8	3	4

Bar width = age interval. Bar height = number of people.

(ii) Median from a histogram

Total = $5 + 8 + 3 + 4 = 20$ people.

Median position $= \dfrac{n+1}{2}$.

$\dfrac{20+1}{2} = \dfrac{21}{2} = 10.5$

So the median sits between the 10th and 11th person.

Count bars left to right: $5$ in $0$–$2$, then the next $8$ are in $2$–$4$.

The 10th and 11th both fall in the $2$–$4$ interval.

The median age lies between $2$ and $4$.

Must learn

1.Bar height = frequency.

2.Bars touch — data is continuous.

3.Median position $= \dfrac{n+1}{2}$. Count bars to find the interval.

YOU TRY · 1

From the same histogram, in which age interval does the median lie?

Median position = $(n+1)/2$. Then count bars.

$n = 20$, position $= 21/2 = 10.5$

5 in 0–2, next 8 in 2–4 — so 10th & 11th sit in 2–4.

Median is in the $2$–$4$ interval.

$2$–$4$ interval

Section 2 of 6

Scatter plot

A scatter plot shows pairs $(x, y)$ as dots. We're looking for a relationship between the two variables.

(i) Example — education vs income

Years of education ($x$) and income in €1,000s ($y$):

Years	Income (€000s)
11	65
11	28
12	30
13	35
13	43
14	55
15	38
16	45
17	58
19	70

(ii) The outlier

One point sticks out — far from the trend:

$(11, 65)$ — the "big farmer".

11 years of education but earning €65,000 — doesn't fit the trend. We flag it and explain.

(iii) Is there a relationship?

Yes. There is a correlation. Positive.

As years of education increase, in general so does income.

Correlation

1.Correlation coefficient is $r$, with $-1 \le r \le 1$.

2.Closer to $1$ or $-1$, the stronger the relationship.

3.Correlation is the LINEAR relationship between $2$ variables.

4.Slope of line of best fit = rate of change of one variable as the other changes.

$r = 0.45 \quad \Rightarrow \quad$ weak positive relationship.

(iv) Using the line of best fit

Equation of the line:

$y = a + bx$

$y = 9.1 + 2.5x$

(1) What is income after 18 years of education?

$x = 18$

$y = 9.1 + 2.5(18) = 54.1$

Income $\approx$ €$54{,}100$

(2) How many years to earn €50,000?

$y = 50$

$50 = 9.1 + 2.5x$

$2.5x = 40.9$

$x = 16.3$ years

(v) Meaning of the slope

$b = m = 2.5$

For every extra year in education, income increases by €2,500.

(vi) Line of best fit without a calculator

Pick any 2 points on the line. Carl uses $(12, 30)$ and $(17, 58)$.

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{58 - 30}{17 - 12} = \dfrac{28}{5}$

$y - y_1 = m(x - x_1)$

$y - 30 = \dfrac{28}{5}(x - 12)$

YOU TRY · 2

Using $y = 9.1 + 2.5x$, predict the income for someone with $20$ years of education.

Sub $x = 20$ into the equation.

$y = 9.1 + 2.5(20) = 9.1 + 50 = 59.1$

$\approx$ €$59{,}100$

Section 3 of 6

Causality

Correlation is not the same as causation.

Causality

Q.Has one variable caused the other to happen?

(i) Smoking causes cancer.

Yes — causation.

(ii) Hot weather causes more ice cream sales.

No — that's only correlation.

Both rise together in summer, but the temperature doesn't cause the sale. People choose to buy.

YOU TRY · 3

A study finds that students who eat breakfast score higher in exams. Does breakfast cause the higher score?

Correlation ≠ causation. Could something else explain it?

It's a correlation. Other factors — sleep, home support, organisation — could be the real cause.

No — correlation only.

No — correlation, not causation.

Section 4 of 6

Stem and leaf

A stem and leaf plot shows the raw data, sorted, with the leaf = one digit only.

The KEY is key — without it, nobody knows what the numbers mean.

(i) Worked example

Draw a stem and leaf of: $143, \; 137, \; 129, \; 133, \; 144$

12	9
13	3 7
14	3 4

Key: $12 \mid 9 = 129$

(ii) Back-to-back stem and leaf

Compare two sets on the same stem. Read the left side outwards.

John: $33, \; 45, \; 37, \; 42, \; 29$

Ann: $29, \; 21, \; 33, \; 48, \; 49$

John		Ann
9	2	1 9
7 3	3	3
5 2	4	8 9

Keys: $9 \mid 2 = 29$ (John) · $4 \mid 9 = 49$ (Ann)

(iii) Compare the two

Compare using mean and standard deviation.

Smaller $\sigma$ = more consistent.

Must learn

1.Leaf = one digit only.

2.The key is key. Always include it.

3.Back-to-back: left side reads outwards.

4.Compare with mean and $\sigma$.

YOU TRY · 4

Looking at the John vs Ann back-to-back plot, which person has the more consistent scores?

Consistent = smaller spread = smaller $\sigma$.

John: $29, 33, 37, 42, 45$ range $= 16$

Ann: $21, 29, 33, 48, 49$ range $= 28$

John — smaller spread, smaller $\sigma$.

John — smaller $\sigma$.

Section 5 of 6

Line plot (dot plot)

A line plot (a.k.a. dot plot) shows frequencies as stacked dots above each value.

(i) Worked example

Goals scored:

Goals	0	1	2	3
Cases	5	3	2	1

Show on a line plot — one × per case, stacked above each value.

The shape on the dot plot tells you about the skew — see next section.

YOU TRY · 5

From the line plot, what is the total number of cases?

Add up all the cases.

$5 + 3 + 2 + 1 = 11$

$11$ cases

$11$

Section 6 of 6

Skew

Skew = the cow's tail. The skew points in the direction of the long tail.

(i) Right skew (positive)

Mode < Median < Mean

Example: family size. Most families small, a few very large pull the mean up.

(ii) Left skew (negative)

Mean < Median < Mode

Example: 6th year shoe size. Most at the upper end, a few smaller pull the mean down.

(iii) Symmetrical

Mode $=$ Mean $=$ Median

Skew — remember

1.Skew = the cow's tail.

2.Right (positive): Mode < Median < Mean. (family size)

3.Left (negative): Mean < Median < Mode. (6th yr shoe size)

4.Symmetrical: all three equal.

YOU TRY · 6

A distribution has Mean $= 8$, Median $= 6$, Mode $= 5$. What's the skew?

Compare the three. Where does the tail point?

Mode $= 5$ < Median $= 6$ < Mean $= 8$. Tail points right.

Right skew (positive).

Right skew (positive)

SUM

The lot in one box

Presenting data toolkit

1.Histogram: continuous data, bars touch. Median position $= (n+1)/2$.

2.Scatter: outlier? relationship? line of best fit $y = a + bx$. Slope $b$ = rate of change.

3.Correlation $r$: $-1 \le r \le 1$. Closer to $\pm 1$ = stronger. LINEAR only.

4.Causality: has one variable caused the other? Correlation ≠ causation.

5.Stem & leaf: leaf $=$ one digit, key is key. Back-to-back reads outwards on left.

6.Line plot: stacked $\times$ above each value.

7.Skew: the cow's tail. Right: Mo < Me < Mean. Left: Mean < Me < Mo.

End of lesson

Presenting Data — HL · Mathslive.ie