1. Example: Suppose these data are recorded for the temper- ature (in ºF) of fifty cities in Luzon in September 2014. 112100127120134118105110109112 110118117116118122114 105109 107112114115118117118122106110 116108110121113120119111104111 120113120117105110118112114 Make a frequency distribution with 7 classes and with 100 as the initial lower class limit.

2. How again? 1 st, compute the range:range = 134 – 100 = 34 2 nd, compute the class width:c.w. = 34/7 = 4.9  5 3 rd, Add the class width to the initial lower limit as many times as the number of classes. These will give us all the lower class limits of the freq. dist. (The tally column is optional.) Class limitsClass BoundariesFrequency 100 - 105 - 110 - 115 - 120 - 125 - 130 -

3. 4 th, to find the upper class limit paired to a lower class limit, just subtract one from the next lower class limit. Class limitsClass BoundariesFrequency 100 – 104 105 – 109 110 – 114 115 – 119 120 – 124 125 – 129 130 - 134

4. 5 th, find the class boundaries, and count the tallies in the frequency column Class limitsClass BoundariesFrequency 100 – 10499.5 – 104.52 105 – 109104.5 – 109.58 110 – 114109.5 – 114.518 115 – 119114.5 – 119.513 120 – 124119.5 – 124.57 125 – 129124.5 – 129.51 130 - 134129.5 – 134.51

5. We can present the grouped freq. dist. in graphical format. Each of the classes are represented by a rectangular bar whose width (on the x-axis) are the boundaries of that class and whose height (on the y-axis) is the corresponding frequency. This diagram is called a HISTOGRAM.

6. RELATIVE FREQUENCY HISTOGRAMS The relative frequencies represents the (tally) frequencies as percentages (%) of the total number of raw data. Class limits Class Boundaries Frequency Relative Frequency 100 – 10499.5 – 104.52 105 – 109104.5 – 109.58 110 – 114109.5 – 114.518 115 – 119114.5 – 119.513 120 – 124119.5 – 124.57 125 – 129124.5 – 129.51 130 - 134129.5 – 134.51 0.40 0.16 0.36 0.26 0.14 0.02 Example: Previously, we had: We use rela- tive frequencies (and its histogram) when the total number of raw data is too large.

7. Nevertheless, the two histograms for a grouped frequency distribution look exactly the same. The only difference in this kind of histogram is that the rela- tive frequencies are placed on the y-axis instead of the tally frequencies.

8. DISTRIBUTION SHAPES The histogram is the most important diagram in Statistics. The histogram is drawn on an xy-plane. The data values (grouped within class boundaries) are placed on the x-axis; and the freq- uencies on the y-axis. DATA VALUES low values high values FREQUENCIES few many

9. Histograms assume peculiar shapes. The most common ones are: ClassesFreq. 60.00 – 61.852 61.85 – 63.706 63.70 – 65.5510 65.55 – 67.4014 67.40 – 69.259 69.25 – 71.107 71.10 – 72.952 Heights of 50 18-y.o. males (in inches) BELL-SHAPED Here, the data appears equally distributed on both sides of the middle (or “average”) values. The tallies increase as data approach the middle values, and then decrease from there on.

10. ClassesFreq. 195 2107 3102 499 594 6103 Outcomes of a die toss (600 times) UNIFORM Here, the data appears equally distributed throughout. The tallies appear constant throughout.

11. Final grades (%) of 100 students in Math 1 RIGHT-SKEWED Here, the tallies accumulate on the lower data values and decrease as the values get higher. ClassesFreq. 60.00 – 65.00 18 65.00 – 70.00 24 70.00 – 75.00 22 75.00 – 80.00 15 80.00 – 85.00 11 85.00 – 95.00 6 95.00 – 100.00 3

12. Final grades (%) of 100 students in P.E. LEFT-SKEWED Here, the tallies accumulate on the higher data values and decrease as the values get lower. (The direction of skewness is on the side of the longer “tail.”) ClassesFreq. 60.00 – 65.002 65.00 – 70.003 70.00 – 75.005 75.00 – 80.009 80.00 – 85.0016 85.00 – 95.0022 95.00 – 100.0025

13. Weights (kgs) of 200 18-y.o. males and females ClassesFreq. 50.00 – 53.005 53.00 – 56.009 56.00 – 59.0018 59.00 – 62.0026 62.00 – 65.0025 65.00 – 68.0013 68.00 – 71.0010 71.00 – 74.0014 74.00 – 77.0020 77.00 – 80.0024 80.00 – 81.0018 81.00 – 84.0010 84.00 – 87.005 87.00 – 90.003 MULTIMODAL Here, the tallies accumulate on more than one data class.

14. FREQUENCY POLYGONS We can also use the middle point (or midpoint or class mark) of a class to represent that class. Class limits Class Boundaries MidpointFrequency 100 – 10499.5 – 104.52 105 – 109104.5 – 109.58 110 – 114109.5 – 114.518 115 – 119114.5 – 119.513 120 – 124119.5 – 124.57 125 – 129124.5 – 129.51 130 - 134129.5 – 134.51 Example: 102 107 112 117 122 127 132

15. We can present the frequency distribution in graphical form where each class is represented by its midpoint. This diagram is called a FREQUENCY POLYGON. Each of the classes is represented by its midpoint on the x- axis, with its frequency as the y-coordinate. The points are then connected by line segments.

16. BAR GRAPHS When the data are qualitative or categorical, bar graphs and pie graphs can be used for presentation. Type of OperationNumber of Cases Thoracic20 Bones and joints45 Eye, ear, nose and throat58 General98 Abdominal115 Urologic74 Proctologic65 Neurosurgery23 Example: Operations performed at PGH (2010)

18. Each of the categorical classes is represented by a vertical or horizontal bar, whose length is its corresponding frequency. This diagram is called a BAR GRAPH. Histograms are bar graphs for quantitative data.

19. PIE GRAPHS Qualitative data can also be presented as sections or sectors of a circle, especially if partitions of a whole has to be emphasized. Example: Operations performed at PGH (2010) Type of OperationNumber of Cases Thoracic20 Bones and joints45 Eye, ear, nose and throat58 General98 Abdominal115 Urologic74 Proctologic65 Neurosurgery23

20. Type of Operation Number of Cases Relative Frequency Sector Angle (º) Thoracic20 0.04 Bones and joints 45 0.09 Eye, ear, nose and throat 58 0.12 General98 0.20 Abdominal115 0.23 Urologic74 0.15 Proctologic65 0.13 Neurosurgery23 0.05 In making pie graphs, the sector angle for each of the classes must be computed. 14º 33º 42º 71º 83º 53º 47º 17º

21. Each of the categorical classes is represented by a sector whose central angle is the sector angle corresponding to its frequency. This diagram is called a PIE GRAPH.