9.04 Review: measures of center and spread

The mean is the average of the values in the data set. It is a measure of center, meaning it is an approximation of where the middle of a data set is.

Let's think about a situation where three friends are planning a trip to Palm Springs. They plan to fly there, and learn that the airline has a rule: each person can only bring 35 lbs of stuff in their bags. On the night before the flight they weigh their luggage and find that their luggage weights from this data set: 29,\,32,\,37

One of them has packed too much. They decide to share their luggage around so that they all carry the same amount. How much does each person carry now?

Thinking about it using more mathematical language, we are sharing the total luggage equally among three groups. As a mathematical expression, we find: \dfrac{29+32+37}{3}=\dfrac{98}{3}=32.67

Each person carries 32.67 lbs. This amount is the mean of the data set.

If we replace every number in a numerical data set with the mean, the sum of the numbers in the data set will be the same. To calculate the mean, use the formula: \text{Mean}=\dfrac{\text{Sum of all numbers}}{\text{How many numbers there are}}

Examples

The median is the middle of the data set when ordered least to greatest. It is also a measure of center.

Let's say seven people were asked about their weekly income, and their responses form this data set: \$300,\,\$400,\,\$400,\,\$430,\,\$470,\,\$490,\,\$2900The mean of this data set is \dfrac{\$5390}{7}=\$770, but this amount doesn't represent the data set very well. Six out of seven people earn much less than this.

Instead we can select the median, which is the middle income. We remove the biggest and the smallest incomes to get: \$400,\,\$400,\,\$430,\,\$470,\,\$490

Then the next biggest and the next smallest to get: \$400,\,\$430,\,\$470

Then the next biggest and the next smallest to get: \$430

There is only one number left, and this is the median - so for this data set the median is \$430. This weekly income is much closer to the other scores in the data set, and summarizes the set better.

The median is the number in the middle of a numerical data set.

• If the list has an odd number of data points, the median is the one right in the center.

• If the list has an even number of data points, the median is the number halfway between (or the average of) the two middle ones.

Half the numbers in the list will be bigger than the median, and half will be smaller.

Examples

The mode of a data set is the result with the greatest frequency, or the data value that appears most often in the data set. If there are multiple results that share the greatest frequency then there will be more than one mode.

Yvonne asks 15 of her friends what their favorite color is. She writes down their answers. Here is what she wrote down: \text{Blue, Pink, Blue, Yellow, Green, Pink, Pink, Yellow,}\\ \text{ Green, Blue, Yellow, Pink, Yellow, Pink, Pink}

She then counts the number of colors to see which is the most picked.

The mode of the data is pink.

Examples

The range is a measure of the spread of a data set from the highest value to the lowest.

Two bus drivers, Kenji and Bjorn, track how many passengers board their buses each day for a week. Their results are displayed in this table:

Both data sets have the same median and the same mean, but the sets are quite different. To calculate the range, we start by finding the highest and lowest number of passengers for each driver:

Now we subtract the lowest from the highest to find the difference, which is the range:

Notice how Kenji's range is quite small, at least compared to Bjorn's. We might say that Kenji's route is more predictable and that Bjorn's route is much more variable (is more likely to change).

The range of a numerical data set is the difference between the highest and the lowest data point. \text{Range = Highest data point - Lowest data point}

Examples