Representation of data ----histogram Alpha

Part One : Class boundaries

The heights of 100 students are recorded to the nearest centimeter. Height,h(cm) …. Frequency7913…. Class boundaries are as below: Height,h(cm) …. Frequency7913…. Example :

The mass of 40 students are recorded to the nearest kilogram. Mass,m(kg) …. Frequency7913…. Class boundaries are as below: so the table are rewrote as below Height,h(cm)45≤m<5555≤m<6565≤m<75…. Frequency7913…. If there is no gap between each class, then what are the boundaries?

Special case1--Age A group of 40 motorists was asked to state the ages at which they passed their driving tests. Age,a (year) Frequency61178 Class boundaries are as below: 17 ， 20 ， 23 ， 26 Age,a (year)17≤a<2020 ≤ a<2323 ≤ a<26a> 26 Frequency61178

Special case 2- Score Score Frequency Score -0.5≤x< ≤x< ≤x< ≤ x < ≤ x<59.5 Fre que ncy Class boundaries are as below: -0.5,9.5,19.5,29.5,39.5,59.5 so we rewrite the table as below when we draw the diagram.

Part two : frequency density

Two important properties of histogram The bars have no spaces between them( though there may be some bars of height zero, which look like spaces) The area of each bar is proportional to the frequency Note: the first one is the main difference between bar chart and histogram

Height, h(cm) frequency 0 ≤ x<53 5 ≤ x< ≤ x< ≤ x< ≤ x< ≤ x< Height, h(cm) If each bar has the same width

Height, h(cm) frequenc y 0 ≤ x<53 5 ≤ x< ≤ x< ≤ x< ≤ x< Height, h(cm) When the widths of bar are unequal and we still take the frequencies as the height of bars. A diagram is drawn as below: Then why this diagram is misleading? frequency

Height, h(cm)frequen cy 0≤ x<53 5 ≤x< ≤ x< ≤x< ≤x< Height, h(cm) Frequency density

So we get:

Now let us draw a histogram

Mass (kg)Frequency 47—544 55—627 63—668 67—747 75—828 83—904 Example : The grouped frequency distribution in Table 1.17 represents the masses in kilograms of a sample of 38 of the people from the datafile ”Brain size”. Represent these data in a histogram

Mass, m(kg) Class boundaries Class width frequen cy Frequenc y density 47— ≤x< — ≤x< — ≤x< — ≤x< — ≤x< — ≤x< First, draw a table and calculate the boundaries and frequency densities.

Mass, m(kg) 1 2 Frequency density Label the two axes with given labels The height of each bars should be correct The boundaries should be right The scale should be and The frequency density should be given.

What if the last class is open-ended? Example: the grouped frequency distribution in the table below summarizes the mass in grams (g), measured to the nearest gram, of sample of 20 pebbles. Represent the data in a histogram Mass (g)Frequency 101— — — — —1502 Over 1504

Mass, m(kg) Class boundariesClass width frequencyFrequency density 101— ≤x< — ≤x< — ≤ x< — ≤x< — ≤x< Over ≤ x< First, draw a table and calculate the boundaries and frequency densities.

Mass, m(kg) Frequency density 0.7 Open-ended interval Take the width of the last interval be the twice that of the previous one

What if the first class is open? What are the advantages and disadvantages of the histogram? What are the advantages and disadvantages of the stem and leaf diagram?

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