1. Representation of data ----histogram Alpha

2. Part One : Class boundaries

3. The heights of 100 students are recorded to the nearest centimeter. Height,h(cm)160-164165-169170-174…. Frequency7913…. Class boundaries are as below: Height,h(cm)159.5-164.5164.5-169.5169.5-174.5…. Frequency7913…. Example :

4. The mass of 40 students are recorded to the nearest kilogram. Mass,m(kg)45-5555-6565-75…. Frequency7913…. Class boundaries are as below: 45 55 65 75 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?

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

6. Special case 2- Score Score0-910-1920-3940-4950-59 Frequency149935 Score -0.5≤x<9.5 9.5≤x<19.519.5≤x<39.539.5≤ x <49.549.5≤ x<59.5 Fre que ncy 149935 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.

7. Part two : frequency density

8. 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

9. Height, h(cm) frequency 0 ≤ x<53 5 ≤ x<105 10 ≤ x<1511 15 ≤ x<206 20 ≤ x<253 25 ≤ x<302 010 5 20 10 30 Height, h(cm) If each bar has the same width

10. Height, h(cm) frequenc y 0 ≤ x<53 5 ≤ x<105 10 ≤ x<1511 15 ≤ x<206 20 ≤ x<305 10 0 5 20 30 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

11. Height, h(cm)frequen cy 0≤ x<53 5 ≤x<105 10 ≤ x<1511 15 ≤x<206 20 ≤x<305 010 5 20 10 30 Height, h(cm) Frequency density

12. So we get:

13. Now let us draw a histogram

14. 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

15. Mass, m(kg) Class boundaries Class width frequen cy Frequenc y density 47—5446.5 ≤x<54.5840.5 55—6254.5 ≤x<62.5870.875 63—6662.5 ≤x<66.5482 67—7466.5 ≤x<74.5870.875 75—8274.5 ≤x<82.5881 83—9082.5 ≤x<90.5840.5 First, draw a table and calculate the boundaries and frequency densities.

16. 5060709080 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 40----95 and 0---2 The frequency density should be given.

17. 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—1101 111—1204 121—1303 131—1407 140—1502 Over 1504

18. Mass, m(kg) Class boundariesClass width frequencyFrequency density 101—110100.5 ≤x<110.51010.1 111—120110.5 ≤x<120.51040.4 121—130120.5≤ x<130.51020.2 131—140130.5 ≤x<140.51070.7 141—150140.5 ≤x<150.51020.2 Over 150150.5≤ x<170.52040.2 First, draw a table and calculate the boundaries and frequency densities.

19. 100180120 160 140 Mass, m(kg) 0.2 0.4 Frequency density 0.7 Open-ended interval Take the width of the last interval be the twice that of the previous one

20. 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?

21. End