Histograms Flashcards

All 26 cards in this deck

What is frequency density?

Frequency $\div$ class width.

How do you work out the class width of a group?

e.g. $20 < w \le 50$

Upper bound $-$ lower bound.

e.g. $50 - 20 = 30$

How do you get the frequency of a class from its frequency density?

e.g. frequency density $2.5$, class $70 < w \le 90$

Frequency $=$ frequency density $\times$ class width.

e.g. $2.5 \times 20 = 50$

What does the area of a bar in a histogram represent?

The frequency of that class.

True or false? In a histogram, the height of each bar shows the frequency of that class.

False. The height is the frequency density; the area of the bar shows the frequency.

True or false? A frequency density must be a whole number.

False. It is frequency $\div$ class width, so it can be a decimal, e.g. $50 \div 20 = 2.5$.

What are the steps to draw a histogram from a grouped frequency table?

e.g. $0 < t \le 10$ has frequency $20$

1. Find each class width: $10 - 0 = 10$
2. Frequency $\div$ class width $=$ frequency density: $20 \div 10 = 2$
3. Draw a bar from $0$ to $10$ of height $2$, axis labelled 'frequency density', with no gaps between bars.

What must the vertical axis of a histogram be labelled?

Frequency density.

In a histogram, what does the width of each bar represent?

The class interval, drawn from its lower bound to its upper bound.

How do you find the height of a missing bar when completing a histogram?

e.g. the class $40 < t \le 60$ has frequency $24$

Divide the frequency by the class width to get the frequency density.

e.g. $24 \div 20 = 1.2$

True or false? A histogram is drawn with gaps between the bars.

False. The data is continuous, so the bars touch each other.

True or false? All the bars of a histogram must be the same width.

False. Unequal class intervals give bars of different widths; the frequency density scale keeps the areas correct.

How do you read the frequency of a class from a histogram?

e.g. a bar of height $1.5$ over $10 < t \le 20$

Frequency density $\times$ class width (the area of the bar).

e.g. $1.5 \times 10 = 15$

How do you estimate the frequency in part of a class interval?

e.g. $150 < h \le 180$ has frequency $18$; estimate for $150 < h \le 160$

Assume the data is spread evenly and take that fraction of the class frequency.

e.g. $\frac{10}{30} \times 18 = 6$

In a histogram, what does one bar having a greater frequency density than another tell you?

That interval contains more values per unit of the quantity, so the data is more concentrated there.

What are the steps to estimate the median from a histogram?

e.g. a histogram showing $43$ runners' times

1. Find the total frequency: $43$
2. Find the median position: $43 \div 2 = 21.5$
3. Add bar frequencies until you reach it, then use proportion across that bar: e.g. $16 + \frac{9.5}{16} \times 4 = 18.375$

For $n$ values shown in a histogram, which position do you use to estimate the median?

The $\frac{n}{2}$ th value (using $\frac{n+1}{2}$ is also accepted).

For $n$ values shown in a histogram, which positions do you use to estimate the lower and upper quartiles?

The $\frac{n}{4}$ th and $\frac{3n}{4}$ th values.

e.g. for $n = 23$: $5.75$ and $17.25$

What is the formula for the interquartile range?

$\text{IQR} = \text{upper quartile} - \text{lower quartile}$

How do you find the median value once you know which bar it lies in?

e.g. $10$ more values needed into a bar of frequency $16$ covering $16 < t \le 20$

Go that fraction of the way across the interval and add it to the lower bound.

e.g. $16 + \frac{10}{16} \times 4 = 18.5$

True or false? The median of data in a histogram is the midpoint of the interval with the tallest bar.

False. You find the interval containing the $\frac{n}{2}$ th value using the bar areas, then use proportion within it.

True or false? If all the class intervals in a histogram have the same width, the heights of the bars are proportional to the frequencies.

True. Each frequency is divided by the same class width, so the frequency densities are in the same proportion as the frequencies.

What are the steps to estimate the fraction of the data in a given range from a histogram?

e.g. $150$ students in total; for $150 < h \le 170$ the whole bars have frequencies $30$, $51$ and $36$, and one third of a bar of frequency $18$ lies in the range

1. Add the frequencies of the whole bars in the range: $30 + 51 + 36 = 117$
2. Add the proportional part of any part-bar: $\frac{1}{3} \times 18 = 6$, giving $123$
3. Write this over the total frequency: $\frac{123}{150}$

True or false? To estimate the proportion of the data in a given range of a histogram, you divide the number of bars in that range by the total number of bars.

False. You divide the frequency (area) in the range by the total frequency (total area).

What are the steps to estimate a percentage of the data in part of a histogram?

e.g. over $50$ years the bars have frequency density $1.4$ for a width of $10$ and $0.7$ for a width of $30$; $20\%$ of these members are female

1. Find each frequency as area: $1.4 \times 10 = 14$ and $0.7 \times 30 = 21$
2. Add them: $14 + 21 = 35$ members over $50$
3. Take the percentage: $20\%$ of $35 = 7$ female members

What are the steps to estimate the value needed to be in the top $20\%$ of a histogram?

e.g. total frequency $60$; the highest bar covers scores $30$ to $40$ and has frequency $12$

1. Find the number needed: $20\%$ of $60 = 12$ people
2. Work down from the highest values, adding frequencies until you reach $12$: the $30$–$40$ bar alone gives $12$
3. Read off the value at that point: a score of $30$