# Calculation of Central Tendencies

Measures of *central tendency* describe the location of the center of a frequency distribution. There are three different measures of central tendency: the *mode, median* and *mean*.

The mode is simply the value of the observation that occurs most frequently. It is useful when you want the prevailing or most popular characteristic or quality. In a survey of adults aged 18 or older, the question "What is your age?" was answered as follows:

## What is your age?

Frequency | Percent | Valid Percent | Cumulative Percent | ||

Valid | 18 | 17 | 1.1 | 1.1 | 1.1 |

19 | 14 | 0.9 | 0.9 | 2.1 | |

20 | 12 | 0.8 | 0.8 | 2.9 | |

21 | 21 | 1.4 | 1.4 | 4.3 | |

22 | 14 | 0.9 | 0.9 | 5.2 | |

23 | 24 | 1.6 | 1.6 | 6.8 | |

24 | 13 | 0.9 | 0.9 | 7.7 | |

25 | 25 | 1.7 | 1.7 | 9.3 | |

26 | 21 | 1.4 | 1.4 | 10.7 | |

27 | 23 | 1.5 | 1.5 | 12.3 | |

28 | 21 | 1.4 | 1.4 | 13.7 | |

29 | 20 | 1.3 | 1.3 | 15 | |

30 | 27 | 1.8 | 1.8 | 16.8 | |

31 | 20 | 1.3 | 1.3 | 18.1 | |

32 | 20 | 1.3 | 1.3 | 19.5 | |

33 | 24 | 1.6 | 1.6 | 21.1 | |

34 | 25 | 1.7 | 1.7 | 22.7 | |

35 | 27 | 1.8 | 1.8 | 24.5 | |

36 | 27 | 1.8 | 1.8 | 26.3 | |

37 | 19 | 1.3 | 1.3 | 27.6 | |

38 | 36 | 2.4 | 2.4 | 30 | |

39 | 32 | 2.1 | 2.1 | 32.1 | |

40 | 33 | 2.2 | 2.2 | 34.3 | |

41 | 31 | 2.1 | 2.1 | 36.4 | |

42 | 42 | 2.8 | 2.8 | 39.2 | |

43 | 32 | 2.1 | 2.1 | 41.3 | |

44 | 39 | 2.6 | 2.6 | 43.9 | |

45 | 34 | 2.3 | 2.3 | 46.2 | |

46 | 47 | 3.1 | 3.1 | 49.3 | |

47 | 30 | 2 | 2 | 51.3 | |

48 | 39 | 2.6 | 2.6 | 53.9 | |

49 | 33 | 2.2 | 2.2 | 56.1 | |

50 | 40 | 2.7 | 2.7 | 58.8 | |

51 | 27 | 1.8 | 1.8 | 60.6 | |

52 | 39 | 2.6 | 2.6 | 63.2 | |

53 | 31 | 2.1 | 2.1 | 65.3 | |

54 | 24 | 1.6 | 1.6 | 66.9 | |

55 | 38 | 2.5 | 2.5 | 69.4 | |

56 | 25 | 1.7 | 1.7 | 71.1 | |

57 | 22 | 1.5 | 1.5 | 72.5 | |

58 | 24 | 1.6 | 1.6 | 74.1 | |

59 | 21 | 1.4 | 1.4 | 75.5 | |

60 | 43 | 2.9 | 2.9 | 78.4 | |

61 | 29 | 1.9 | 1.9 | 80.3 | |

62 | 28 | 1.9 | 1.9 | 82.2 | |

63 | 25 | 1.7 | 1.7 | 83.9 | |

64 | 19 | 1.3 | 1.3 | 85.1 | |

65 | 39 | 2.6 | 2.6 | 87.7 | |

66 | 20 | 1.3 | 1.3 | 89.1 | |

67 | 14 | 0.9 | 0.9 | 90 | |

68 | 22 | 1.5 | 1.5 | 91.5 | |

69 | 25 | 1.7 | 1.7 | 93.1 | |

70 | 19 | 1.3 | 1.3 | 94.4 | |

71 | 9 | 0.6 | 0.6 | 95 | |

72 | 20 | 1.3 | 1.3 | 96.3 | |

73 | 8 | 0.5 | 0.5 | 96.9 | |

74 | 16 | 1.1 | 1.1 | 97.9 | |

75 | 10 | 0.7 | 0.7 | 98.6 | |

76 | 4 | 0.3 | 0.3 | 98.9 | |

77 | 4 | 0.3 | 0.3 | 99.1 | |

78 | 4 | 0.3 | 0.3 | 99.4 | |

79 | 1 | 0.1 | 0.1 | 99.5 | |

80 | 4 | 0.3 | 0.3 | 99.7 | |

81 | 1 | 0.1 | 0.1 | 99.8 | |

82 | 1 | 0.1 | 0.1 | 99.9 | |

84 | 1 | 0.1 | 0.1 | 99.9 | |

86 | 1 | 0.1 | 0.1 | 100 | |

Total | 1500 | 100 | 100 |

Hence the mode is 46, since 47 respondents provided this answer, more than any other category. Since there is only one mode in this distribution, it is referred to as ‘unimodal’. If a distribution has two modes (or two values that have the same amount of responses that are also the highest), it is referred to as ‘bimodal’.

The median is the middle observation, where half the respondents have provided smaller values, and half larger ones. It is calculated by arranging all observations from lowest to highest score and counting to the middle value. In our example above, the median is 47. The cumulative percentage will tell you at a glance where the median falls. Since the median is not as sensitive as the mean to extreme values, it is used most commonly in cases where you are dealing with ‘outliers’ or extreme values in the distribution that would skew your data in some way. The median is also useful when dealing with ordinal data and you are most concerned with a typical score.

The mean is also known as the ‘arithmetic average’ and is symbolized by ‘X’. The formula for calculating the mean is ∑ X/n (The Greek letter sigma ∑ is the symbol for sum) This means that you total all responses (X) and then divide them by the total number of observations (n). In our example, you would have to multiply all the values (actual ages or ‘x’) by the number of respondents or frequencies (‘f’) for each (∑ x • f = X or. 18 • 17 + 19 • 14 + etc.) and then divide the total by the 1500 respondents who participated in the survey. The result is 47.04. Since the computer will follow the same steps, you must be sure that the values are real and not just codes for categories. For example, the computer would calculate a mean of 3.7367 for the same information as the frequency above but recoded into age categories, based on the assumption that the values under x are 1 to 6:

## What is your age?

Frequency | Percent | Valid Percent | Cumulative Percent | ||

Valid | 18-24 years (1) | 115 | 7.7 | 7.7 | 7.7 |

25-34 years (2) | 226 | 15.1 | 15.1 | 22.7 | |

35-44 years (3) | 318 | 21.2 | 21.2 | 43.9 | |

45-54 years (4) | 344 | 22.9 | 22.9 | 66.9 | |

55-64 years (5) | 274 | 18.3 | 18.3 | 85.1 | |

65 years or more (6) | 223 | 14.9 | 14.9 | 100 | |

Total | 1500 | 100 | 100 |

Another useful calculation is the range, the calculation of the spread of the numerical data. It is calculated by subtracting the lowest value (in our example 18) from the highest value (or 86) to give us a total of 68. This is particularly useful when dealing with rating scores, for instance, where you would like to determine how close people are in agreement or alternatively, how wide the discrepancies are.