TYPES OF SUMMARY MEASURES

Measures of Central Tendency:

• Mean, Median, Mode

• Standard Deviation

µ = (x₁ + x₂ + .. + x_N)/N

    = S x_i /N

    = (1.34 + 0.45 + … + 1.29)/25 = 32.93/25

    = 1.32, based on data in table 2.1 or 1.8.

Summation sign S means that the numbers to the right of the summation sign (i.e., the values of x_i) should be summed from the lower limit on i (which is given below the S sign) to the upper limit on i (which is given above the S sign).
 

Arithmetic Average Based on Grouped Data:

Very often raw data is not available, but on grouped data or frequency distribution is. So we can not use the above formula to calculate mean. But we can approximate S x_i by assuming that the midpoint of each class interval can be used to represent the value of the measurements in that class. Consider the frequency distribution of the raw data in table 1.8 or table 2.1 given in Table 1.9, page 17:

Class interval     Mid point     Frequency

0.20 - 0.60          0.40             5

0.60 – 1.00         0.80             3

1.00 - 1.40         1.20             8

1.40 - 1.80         1.60             4

1.80 - 2.20         2.00             3

2.20 - 2.60         2.40             1

2.60 - 3.00         2.80             0

3.00 - 3.40         3.20             1
______________________________

Total                                    25

X_bar = (f₁.x₁ + f₂.x₂ + … + f_k.X_k)/n

Note S f_i = n.

In our example,

X_bar = {0.4 (5) + 0.8 (3) + 1.2 (8) + 1.6 (4) + 2.0 (3)

            + 2.4 (1) + 2.8 (0) + 3.2 (1)}/25

        = 32/25 =1.28.

Based on ungrouped data we got 32.93/25 = 1.32.

Weighted Arithmetic Mean:

Example: Mean profit rate for a group of firms. Profit rates of different firms should be weighted according to their size.

X_w = ( S w_ix_i)/ S w_i

Example: Suppose 3 firms with profit rates 10%, 12% and 15%. But their sizes are 2 billion, 1 billion and 1 billion. Then weighted average of the profit rate will be: 2x10 + 1X12 + 1x15 = 47/4 = 11.75.

Another good example is the construction of inflation rate. Different commodities will have different inflation rates, but their importance is not the same for consumers. Housing prices, food, and medical care are more important than pencils, tea, etc.
 

MEDIAN

It is the middle most value of the relevant set of data.

That is, the median is the value that divides the set of data in half, 50% of the measurements being above (or equal to) it and 50% being below it. If we order the values, it will be in the middle with odd number of observations. In our example the 13^th value is 1.32.

With even number of observations, take the average of the two middle observations.

Example, 2,4, 6, 8 – median is 7

Mode

Mode, another measure of central tendency, is defined as the most frequently observed value of the measurements in the data set. Out of the 70 students in the class, suppose following is the number of students wearing shirts of different color:

Red: 10
White: 30 -------------- The modal value is White
Blue: 15
Green: 7
Orange: 8
Total: 70

Another example from the Text: Out of 19 families in an apartment complex, 8 families earn $30,000, 10 earn $35,000, and 1earns $1 million. The mode is $30,000 because it is the most frequently observed value in the sample.

Frequency polygon can easily detect the modal value: It is the value along the horizontal axis (called abscissa) where the frequency polygon achieves its maximum vertical axis.

Mode
of
a frequency Figure 2.1 page 48
polygon

Some frequency distributions have more than one mode.
Bimodal distributions have two modes.
There are multimodal distributions.

Example of a bimodal distribution: Frequency distribution of American adults. Males and females will have two different modes. When distributions are multimodal, one has to be careful about measures like mean or median. They may fall in between two modes, and can be unrepresentative of the bulk of the data lying near the separate modes.

Frequency
polygon
of a bimodal Figure 2.2, Page 49
frequency
distribution.

Relationship between Mean, Median and Mode:

For a symmetric unimodal distribution, mean, median, and mode will be identical.

Many distributions are skewed to the left or to the right. A frequency distribution that is skewed to the right has a long tail to the right, whereas one that is skewed to the left has a long tail to the left.

Frequency distribution of income or asset holdings of individuals is generally skewed to the right because there are always a few who earn a lot.

Can you think of a frequency distribution that is skewed to the left?

For asymmetric distributions, median falls between mean and mode.

 Look at Figure 2.3 of the Text (page 2.3).

Mean Vs. Median

• Sensitivity to extreme values – median is better
• Open-ended class intervals – median is better
• Mathematical Convenience – mean is better
• Extent of sampling variability – mean is better