Histograms, Grouped Data & Std Dev

Section 1 of 3

Histograms

For continuous data

A histogram is only for numerical continuous data in intervals. The bars touch (no gaps). $4\text{-}8$ means ‘$4$ up to but not including $8$’.

Ages $0\text{-}4$: $8$, $4\text{-}8$: $7$, $8\text{-}12$: $6$, $12\text{-}16$: $9$:

You try

Draw a histogram for ages $0\text{-}10$: $7$, $10\text{-}20$: $9$, $20\text{-}30$: $3$, $30\text{-}40$: $8$, $40\text{-}50$: $4$, $50\text{-}60$: $2$. What type of data, and the modal interval?

Continuous (age on a scale); modal interval = tallest bar.

Numerical continuous

Modal interval $= 10\text{-}20$

continuous; modal $10\text{-}20$

Section 2 of 3

The Mean of Grouped Data

Use the mid-interval

You don’t know the exact values, so use the mid-point of each interval: $0\text{-}2 \to 1$, $2\text{-}4 \to 3$, and so on. Enter those with the frequencies in the calculator’s stats mode for $\bar{x}$ and $\sigma_x$.

Worked example — grouped mean

Ages $0\text{-}2$: $7$, $2\text{-}4$: $9$, $4\text{-}6$: $3$, $6\text{-}8$: $10$, $8\text{-}10$: $9$. Find $\bar{x}$ and $\sigma_x$.

Mid-points $1,3,5,7,9$ with frequencies $7,9,3,10,9$

$\bar{x} = 5.26 \approx 5.3$, $\sigma_x = 2.94 \approx 2.9$

$5.3,\ 2.9$

Worked example — grouped mean

Ages $0\text{-}2$: $15$, $2\text{-}4$: $20$, $4\text{-}6$: $35$, $6\text{-}8$: $40$, $8\text{-}10$: $10$. Find $\bar{x}$ and $\sigma_x$.

Mid-points $1,3,5,7,9$

$\bar{x} = 5.14 \approx 5.1$, $\sigma_x = 2.30 \approx 2.3$

$5.1,\ 2.3$

You try

Ages $0\text{-}10$: $7$, $10\text{-}20$: $9$, $20\text{-}30$: $3$, $30\text{-}40$: $8$, $40\text{-}50$: $4$, $50\text{-}60$: $2$. Find $\bar{x}$ and $\sigma_x$.

Mid-points are $5,15,25,35,45,55$.

$\bar{x} = 24.69 \approx 24.7$, $\sigma_x = 15.5$

$24.7,\ 15.5$

Section 3 of 3

Standard Deviation & Choosing a Chart

Spread, by calculator

$\sigma_x$ measures spread about the mean — the higher $\sigma_x$, the more spread out. Enter each value with its frequency in stats mode and read $\bar{x}$ and $\sigma_x$.

Worked example — from a table

Goals $0$ (×$2$), $1$ (×$12$), $2$ (×$7$), $3$ (×$3$), $4$ (×$9$). Find $\bar{x}$ and $\sigma_x$.

$\bar{x} = 2.15$, $\sigma_x = 1.3$

$2.15,\ 1.3$

Worked example — from a table

Goals $0$ (×$2$), $1$ (×$3$), $2$ (×$7$), $3$ (×$8$), $4$ (×$4$), $5$ (×$1$). Find $\bar{x}$ and $\sigma_x$.

$\bar{x} = 2.48 \approx 2.5$, $\sigma_x = 1.23 \approx 1.2$

$2.5,\ 1.2$

Worked example — from a table

Age $2$ (×$3$), $3$ (×$4$), $4$ (×$13$), $5$ (×$12$), $7$ (×$9$). Find $\bar{x}$ and $\sigma_x$.

$\bar{x} = 4.70 \approx 4.7$, $\sigma_x = 1.46 \approx 1.5$

$4.7,\ 1.5$

You try

Goals $0$ (×$4$), $1$ (×$6$), $2$ (×$2$), $3$ (×$7$), $4$ (×$9$). Find $\bar{x}$ and $\sigma_x$.

Same calculator method.

$\bar{x} = 2.39 \approx 2.4$, $\sigma_x = 1.47 \approx 1.5$

$2.4,\ 1.5$

Which chart?

Line plot / bar chart → categorical or discrete. Histogram → numerical continuous.

That’s Class 6.

Histograms, the mean of grouped data from mid-points, standard deviation, and choosing a chart. Class 7: stem-and-leaf plots.