__Measures of Central Tendency__: Mean & weighted mean, Mode, Median, Midrange.

__Arithmetic Mean__:

**Ungrouped data: Mean = Ʃ x / n **

**Grouped data: Mean = Ʃ f X / n where f refers to the frequency and x refers to the observations or to midpoint classes. **

__Ungrouped data Mean__ = Ʃ x / n

**Example: The ages of 5 patients are : 20 , 45 , 30 , 55 & 70 years , so the mean age of the group will be : ( 20 + 45 + 30 + 55 + 70 ) / 5 = 220 / 5 = 44 years.**

__Grouped data Mean__ ( X ) = Ʃ f X / n where f refers to the frequency and X refers to observations and n refers to the number.

__Example__: 40 individuals, 5 of which having 60 KG weight, 20 having 65 KG weight & 15 having 70 KG weight, so the mean weight will be (

** 5 x 60 ) + ( 20 x 65 ) + ( 15 x 70 ) / 40 = 66.25 KG**

__Grouped data Mean__** ( X ) = Ʃ f X / n where f refers to the frequency and X refers to midpoint classes and n refers to the number**

Simple frequency | Height in cms |

6 | 159 – |

15 | 162 – |

31 | 165 – |

22 | 168 – |

20 | 171 – |

4 | 174 – |

2 | 177 + |

**Looking to the previous table, n = 6 +15+31+22+20+4+2 = 100 and f is the frequency which is 6 , 15 , 31 , 22 , 20 , 4 and 2 **

**There are 7 classes of data and the X which is the midpoint class is required to be obtained. **

**The midpoint class = the mean of 2 adjacent classes. **

**So the 1 ^{st} midpoint class = 159 +162 / 2 = 160.5, by knowing this 1^{st} class, we can obtain the other classes by simply adding 3 to each class (3 is the class length) so the other midpoint classes will be 163.5, 166.5, 169.5, 172.5, 175.5 and 178.5**

f | Midpoint class |

6 | 160.5 |

15 | 163.5 |

31 | 166.5 |

22 | 169.5 |

20 | 172.5 |

4 | 175.5 |

2 | 178.5 |

**By using the equation Mean = Ʃ f X / n, the mean will be **

**{(160.5 x 6) + (163.5 x 15) + (166.5 x 31) + (169.5 x 22) + (172.5 x 20) + (175.5 x4) + (178.5 x 2)} / 100 = 168.15 Cms **

__Weighted Mean__**: Is the mean of the means**** **

**It equals X1 x n1 + X2 x n2 + X3 x n3 + ……………….. / n1 + n2 + n3+**** **

**Weighted mean can be used as a rough estimate of the universe mean. If you don’t know the universe mean, obtain the mean from different surveys and calculate the weighted mean. **

__Mode__: is the most frequent observation.

**Data may have no mode, may be uni – modal, bi- modal or poly- modal. Example: 1, 2, 3, 4, 5 is a no modal set of data while 1, 2, 2, 3, 4, 5 is uni – modal and 1, 1, 2, 2, 3, 4, 5 is bi- modal while 1, 1, 2, 2, 3, 3, 4, 5 is poly- modal. **

__Median__: is the number that bisects the observations into equal values. To obtain the median , arrange the values by order from the highest to lowest or from the lowest to highest then the order of the median value will be :

** ( n + 1 ) / 2 If the data are odd number. ( n / 2 ) & ( n/ 2 ) + 1 If the data are even number. **

__Example(1)__: Estimate the median for the following set of da12, 17, 9, 18, 14, 22, 26

**This set of data is odd number (7 observations) **

**The data are to be arranged: 9, 12, 14, 17, 18, 22, 26 **

**Then the order of the median will be (n + 1) / 2 = (7 + 1) / 2 = 4 **

**So the median for these data is the 4 ^{th} number which is 17 **

__Example(2) __: Estimate the median for the following set of data:

**14, 6, 8, 12, 10, 24 **

**This set of data is even number (6 observations) **

**The data are to be arranged: 6, 8, 10, 12, 14, 24 **

**hen the order of the median will be : (n /2) & (n /2) + 1 = (6/2) & ( 6/2 ) + 1 = 3 & 4 **

**So the median values for these data are the 3 ^{rd} & 4^{th} values, 10 &12 and the median = (10+12 ) / 2 = 11 **

__Midrange__: is the highest plus the lowest values divided by 2.

**Example: Range of normal Hb is 10 → 16 gm **

**Midrange = ( 16 + 10 ) / 2 = 13**

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