Calculate using histogram bar dimensions

Questions where students must use the physical dimensions (height and width) of histogram bars to find frequency density or frequency values, requiring understanding that area represents frequency.

4 questions

CAIE S1 2010 November Q4
4 The weights in grams of a number of stones, measured correct to the nearest gram, are represented in the following table.
Weight (grams)\(1 - 10\)\(11 - 20\)\(21 - 25\)\(26 - 30\)\(31 - 50\)\(51 - 70\)
Frequency\(2 x\)\(4 x\)\(3 x\)\(5 x\)\(4 x\)\(x\)
A histogram is drawn with a scale of 1 cm to 1 unit on the vertical axis, which represents frequency density. The \(1 - 10\) rectangle has height 3 cm .
  1. Calculate the value of \(x\) and the height of the 51-70 rectangle.
  2. Calculate an estimate of the mean weight of the stones.
OCR PURE 2066 Q8
8
  1. Joseph drew a histogram to show information about one Local Authority. He used data from the "Age structure by LA 2011" tab in the large data set. The table shows an extract from the data that he used.
    Age group0 to 4
    Frequency2143
    Joseph used a scale of \(1 \mathrm {~cm} = 1000\) units on the frequency density axis. Calculate the height of the histogram block for the 0 to 4 class.
  2. Magdalene wishes to draw a statistical diagram to illustrate some of the data from the "Method of travel by LA 2011" tab in the large data set. State why she cannot draw a histogram.
Edexcel S1 2009 January Q5
5. In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month. The results are summarised in the table below.
Number of hoursMid-pointFrequency
0-52.7520
6-76.516
8-10918
11-151325
16-2520.515
26-503810
A histogram was drawn and the group ( \(8 - 10\) ) hours was represented by a rectangle that was 1.5 cm wide and 3 cm high.
  1. Calculate the width and height of the rectangle representing the group (16-25) hours.
  2. Use linear interpolation to estimate the median and interquartile range.
  3. Estimate the mean and standard deviation of the number of hours spent shopping.
  4. State, giving a reason, the skewness of these data.
  5. State, giving a reason, which average and measure of dispersion you would recommend to use to summarise these data.
Edexcel S1 2014 June Q6
6. The times, in seconds, spent in a queue at a supermarket by 85 randomly selected customers, are summarised in the table below.
Time (seconds)Number of customers, \(f\)
0-302
30-6010
60-7017
70-8025
80-10025
100-1506
A histogram was drawn to represent these data. The \(30 - 60\) group was represented by a bar of width 1.5 cm and height 1 cm .
  1. Find the width and the height of the \(70 - 80\) group.
  2. Use linear interpolation to estimate the median of this distribution. Given that \(x\) denotes the midpoint of each group in the table and $$\sum f x = 6460 \quad \sum f x ^ { 2 } = 529400$$
  3. calculate an estimate for
    1. the mean,
    2. the standard deviation,
      for the above data. One measure of skewness is given by $$\text { coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
  4. Evaluate this coefficient and comment on the skewness of these data.