1. STATISTICS AND NUMERICAL METHODS MATH 0102 Measures of Central Location NURAZRIN JUPRI1

2. Mean (Ungroup data) NURAZRIN JUPRI2  called the arithmetic mean,  by sharing the sum of the quantities concerned equally between the numbers of quantities.  RULE: add up the data provided and divide by the number of quantities.

3. Example: What is the arithmetic mean of the monthly takings? NURAZRIN JUPRI3 An investigation into the takings of a small grocer’s shop gives the following results: £  January2 794  February1 986  March2 325  April3 654  May3 726  June3 985  July6 574  August7 384  September5 259  October3 265  November4 381  December5 286 £50 619

4. Median (Ungroup data) NURAZRIN JUPRI4  Median: that value which divides the data into two equal halves; 50% of values lying below and 50% above the median.  Array: place data in numerical order – whether rising or falling  Median position is n + 1 2 where n = number of values  Median value is that value which corresponds to the median position

5. The median is the value of the middle item of a distribution once all of the items have been arranged in order of magnitude. The median of the following nine values: 264 24 11 12 28 86 90 2 The median of the following ten values: 264 24 11 12 28 86 90 2 8

6. TRY! NURAZRIN JUPRI6 1. Model A: 1, 2, 18, 23, 26, 42, 294 2. Model B: 43, 44, 45, 69, 73, 76 Find the median?

7. Mode (Ungroup data) NURAZRIN JUPRI7  Mode: that value which occurs most often (i.e. with the highest frequency)  Example: a) 1 2 1 2 3 4 6 1 2 2 7 b) 1 2 1 2 4 1 1 2 2 7 c) 1 2 3 4 9 5 6 8 7

8. Example: NURAZRIN JUPRI8

9. Example: central location for ungrouped data 9  The following data measures the attention span in minutes of 15 undergraduates in a sociology lecture. 4, 6, 7, 8, 8, 8, 8, 9, 9, 10, 11, 12, 14, 15, 18 a) Find the arithmetic mean b) Find the median c) Find the mode

10. Central location: grouped data 10  Grouped data: data which is only available in grouped form e.g. class intervals in frequency table  Class mid-points: we assume that the data in any class interval all fall on the class mid-point. Put another way, the data are equally spread along any given class interval

11. Mean (grouped data)  Where F i = frequency of ith class interval X i = mid-point of ith class interval j = number of class intervals 11 Note: simplifying assumption: all values in a class interval are equally spread along that interval

12. Daily Demand FrequencyMid pointFX 6-108864 11-1541352 16-20618108 18224 Find arithmetic mean of grouped data Find mid-point of each class interval. Arithmetic mean x = ∑ fx / ∑f = 224/ 18 = 12.4units Daily DemandFrequency 6-108 11-154 16-206

13. Median (grouped data) NURAZRIN JUPRI13 LCB = lower class boundary CF = the cumulative number of frequencies in the classes preceding the class containing the median F = the frequency of the median class

14. Mode (grouped data) NURAZRIN JUPRI14  Modal class interval: that class interval in which the mode value falls