Lesson 2.3 Mean, Median and Mode

Mean

The mean is the same as the average. To find the mean, add all the values and divide by the total number of values.

Example: {2, 3, 5, 6}

The mean

The letter x with a bar over it, , represents the sample mean.

The Greek letter, µ , represents the population mean.

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Median

The median is the middle value of a set of numbers that has been ordered from smallest to largest. The upper case letter M is used for the median.

Example: A sample of statistics exam scores for 14 students are (in order from smallest to largest) as follows:

53, 59, 63, 63, 72, 72, 76, 78, 81, 83, 84, 84, 90, 93

Notice that 14 is an even number.

The median is between 7th and 8th values (the middle two values). The 7th value is 76 and the 8th value is 78. The median M =

Example: A second sample of statistics exam scores for 15 students are (in order from smallest to largest) as follows:

52, 60, 65, 67, 70, 71, 74, 76, 78, 78, 78, 80, 86, 89, 95

Notice that 15 is an odd number.

The median is the 8th value (the middle value). The 8th value is 76 so the median M = 76.

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Mode

The mode is the most frequent value in the set of numbers.

Example: In the data set

52, 60, 65, 67, 70, 71, 74, 76, 78, 78, 78, 80, 86, 89, 95, the most frequent value is 78. The mode = 78.

Example: In the data set

52, 53, 53, 53, 60, 67, 72,72,72, 90, both 53 and 72 occur the most number of times (3 times each) so there are two modes, 53 and 72. We call this set of data bimodal meaning it has two modes.

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Think About It

Which is the "best" measure of the center of a set of data? That all depends on the data. If the data tend toward being different values with no extremely large or small values, the mean might be the best measure of the data. If the data is highly skewed, the median might be the best measure of the center. If the data repeats values, the mode might be the best measure.

Example: A shoe store sells women's shoes in sizes 5 through 11. The average shoe size is 7 1/2, the median size is 8 1/2, and most frequent shoe size is 8. Which is the best measure?

Because the most frequent shoe size is 8, the manager orders more shoes in size 8 than any other size. Even though the average shoe size is 7 1/2 and the median is 8 1/2, the mode is the best measure of the center for this particular shoe store.

Example: Suppose in a community of 7 households, the yearly income is (in dollars)

50,000

53,000

58,000

59,000

60,000

61,000

5,000,000

The mean is $763,000 and the median is $59,000. Which is the best measure?

The median better represents the center of income of the majority of households. In this case, the median is the better measure of the center of the data.

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TI-83 Example Showing the Mean, the Median, and the Quartiles

Example

The TI-83/84 easily calculates the mean, the median, and the quartiles.

NOTE: These directions are for entering data with a TI-83 or TI-84 calculator.

Sample Data

We are manipulating 1-variable statistics.

To begin:

Step 1.  Turn on the calculator.

Step 2.  Access statistics mode. (STAT)

Step 3.  Select <4:ClrList> to clear data from lists, if desired.

Step 4.  Enter list 2^nd L1, 2^nd L2 to be cleared. ENTER

Step 7.  Access statistics mode. (STAT)

Step 8.  Select <1:Edit >

Step 9.  Enter data. Data values go into [L1]. (You may need to arrow over to [L1])  Type in a data value and enter it. (For negative numbers, use the negate (-) key at the bottom of

the keypad).  Continue in the same manner until all data values are entered.

Step 10.  In [L2], enter the frequencies for each data value in [L1].  Type in a frequency and enter it. (If a data value appears only once, the frequency is "1")  Continue in the same manner until all frequencies are entered.

Step 11.  Access statistics mode. (STAT)

Step 12.  Navigate to <CALC>

Step 13.  Access <1:1-var Stats>  ENTER

Step 14.   Type 2^nd L1,2^nd L2 ENTER

Step 16.  The statistics should be displayed. You may arrow down to get remaining statistics. Repeat as necessary.

Please continue to the next section of this lesson.

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Up » 2.1 Graph » 2.2 Quartiles and Percentiles » 2.3 Mean, Median and Mode » 2.4 Variance and Standard Deviation