Measures of Central Tendency for Ungrouped Data

Mean Mean = Sum of all Values Number of Values Mean from a sample is x = ∑x/n Mean from population is µ = ∑x/N Mean is very sensitive to outliers.

EXAMPLE - the grade 10 math class recently had a mathematics test and the grades were as follows: 78 66 82 89 75 . + 74 464 464 / 6 = 77.3 Hence, 77.3 is the mean average of the class.

Answer: 89, 73, 84, 91, 87, 77, 94 Solution: Problem: Scott took 7 math tests in one marking period. What is the mean test score?   89,  73,  84,  91,  87,  77,  94 Solution:   The sum of these numbers is 595. Dividing the sum by the number of test scores we get: Answer:   The mean test score is 85.

66 mph, 57 mph, 71 mph, 54 mph, 69 mph, 58 mph Example: Find the mean driving speed for 6 different cars on the same highway.   66 mph,  57 mph,  71 mph,  54 mph,  69 mph,  58 mph Solution:   66 + 57 + 71 + 54 + 69 + 58 = 375 Answer:   The mean driving speed is 62.5 mph.

Median Middle term of data after ranked in increasing order. Divides data into two equal parts. Not influenced by oultiers.

example 66 74 75 78 82 89 - as you can see we have two numbers, there is no middle number. What do we do? It is simple; we take the two middle numbers and find the average, ( or mean ) 75 + 78 = 153 153 / 2 = 76 Hence, the middle number is 76.5.

Mode Value with the highest frequency in a data set. this is the number that occurs most often

example find the mode of the following data: 78  56  68  92  84  76  74  56  68  66  78   72  66 65  53  61  62  78  84  61  90  8 7  77  62  88  81 The mode is  78. 

Mode Not all data sets have a mode. Some data sets have more than one mode. Unimodal – One mode. Bimodal – Two modes. Multimodal – More than two modes.

Comparisons of Measures of Center Mean is most common and each member of the data set is used in its calculation. Median is better if the data set contains outliers. Mode is the easiest to locate, but not much use.

Advantages of mean It is easy to understand & simple calculate. It is based on all the values. It is rigidly defined . It is easy to understand the arithmetic average even if some of the details of the data are lacking. It is not based on the position in the series.

Disadvantages of mean It is affected by extreme values. It cannot be calculated for open end classes. It cannot be located graphically It gives misleading conclusions. It has upward bias.

Advantages of Median: Median can be calculated in all distributions. Median can be understood even by common people. Median can be ascertained even with the extreme items. It can be located graphically It is most useful dealing with qualitative data

Disadvantages of Median It is not based on all the values. It is not capable of further mathematical treatment. It is affected fluctuation of sampling. In case of even no. of values it may not the value from the data.

Advantages of Mode : Mode is readily comprehensible and easily calculated It is the best representative of data It is not at all affected by extreme value. The value of mode can also be determined graphically. It is usually an actual value of an important part of the series.

Disadvantages of Mode : It is not based on all observations. It is not capable of further mathematical manipulation. Mode is affected to a great extent by sampling fluctuations. Choice of grouping has great influence on the value of mode.

CONCLUSION  A measure of central tendency is a measure that tells us where the middle of a bunch of data lies. Mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers in a set of data. This is also known as average.

Median is the number present in the middle when the numbers in a set of data are arranged in ascending or descending order. If the number of numbers in a data set is even, then the median is the mean of the two middle numbers. Mode is the value that occurs most frequently in a set of data.

Mean, median, and mode for a symmetric histogram and frequency distribution curve.

Mean, median, and mode for a histogram and frequency distribution curve skewed to the right.

Mean, median, and mode for a histogram and frequency distribution curve skewed to the left.