Measures of Central Tendency

Calculating the Measures of Central Tendency of Ungrouped Data

Objectives

• Determine the measures of central tendency of ungrouped data.

Measures of Central Tendency

• After collecting data, it is presented through tables and graphs.

• Statistical measures aid in analyzing data.

• A single value must be used to represent the general magnitude and characteristics of the collected data.

• This value must be taken from the center, not the extremes.

• Measures of central tendency are single values that represent the general magnitude or characteristics of a set of data.

• Three measures of central tendency: mean, median, and mode.

• These measures are also called measures of average.

  • Mean: computational average

  • Median: positional average

  • Mode: inspectional average

• Datasets can be:

  • Grouped: presented using a frequency distribution

  • Ungrouped: not presented using a frequency distribution

• This lesson focuses on measures of central tendency for ungrouped data.

Mean of Ungrouped Data

• Denoted by $\bar{x}$ (x-bar) for sample mean and $\mu$ (mu) for population mean.

• Data are assumed as values from samples unless specified otherwise.

• Applicable for data with quantitative values (discrete or continuous) at interval or ratio levels of measurement.

• Formula: $\bar{x} = \frac{\sum x}{N}$

  • $x$: individual score or value

  • $N$: total number of cases

Example 1

Find the mean of each data set.
a. Scores of Xander in six math quizzes: 21, 26, 22, 18, 30, 24
$\bar{x} = \frac{21+26+22+18+30+24}{6} = \frac{141}{6} = 23.5$
The mean score of Xander in six math quizzes is 23.5.
b. Temperatures (in °C) in Baguio City in the previous week: 12.3, 11.7, 13.0, 14.1, 15.2, 10.4, 13.4
$\bar{x} = \frac{12.3+11.7+13+14.1+15.2+10.4+13.4}{7} = \frac{90.1}{7} = 12.87$
The mean temperature in Baguio is approximately 12.87 °C.

Example 2

Buboy sold an average of 13 balut eggs for the past 5 days. If he sold 15 on Monday, 12 on Tuesday, 13 on Wednesday, and 14 on Thursday, how many balut eggs did he sell on Friday?

• Number of balut eggs: $x<em>1 = 15, x</em>2 = 12, x<em>3 = 13, x</em>4 = 14, x_5$

• Average: $\bar{x} = 13$
  $13 = \frac{15+12+13+14+x<em>5}{5}$ $13 = \frac{54+x</em>5}{5}$
  $65 = 54 + x<em>5$ $x</em>5 = 11$
  Therefore, Buboy sold 11 balut eggs on Friday.

Your Turn

1. Find the mean:
   a. 80, 87, 92, 94, 88, 84, 89, 91
   $\bar{x} = \frac{80 + 87 + 92 + 94 + 88 + 84 + 89 + 91}{8} = 88.125$
   b. 21113, 34234
   $\bar{x} = \frac{21113 + 34234}{2} = 27673.5$

2. Yam needs a mean grade of 75 to pass the subject. She got 78, 72, and 71 on the first 3 quarters. What grade must she get in the last quarter to pass?
   $75 = \frac{78 + 72 + 71 + x<em>4}{4}$ $75 = \frac{221 + x</em>4}{4}$
   $300 = 221 + x<em>4$ $x</em>4 = 79$

Median of Ungrouped Data

• Denoted by $\tilde{x}$ (x-tilde).

• Middle value in an ordered set of data.

• Applicable for data in the ordinal, interval, or ratio levels of measurement.

• Steps in finding the median of ungrouped data:

  1. Arrange the values in the data set either in ascending or descending order.

  2. Assign a number $N$ to each arranged value consecutively.

  3. Determine the median:
     a. If $N$ is odd, the median is the $\frac{N+1}{2}$th value.
     b. If $N$ is even, the median is the mean of the $\frac{N}{2}$th and $(\frac{N}{2} + 1)$th values.

Example 3

a. The birth order of seven students: 2nd, 6th, 1st, 2nd, 5th, 9th, 8th
Arranged: 1st, 2nd, 2nd, 5th, 6th, 8th, 9th
Assigned numbers: 1, 2, 3, 4, 5, 6, 7
$N = 7$ (odd), so the median is the $\frac{7+1}{2} = 4$th value.
The 4th value is 5th. Hence, the median birth order is 5th.
b. Daily wages of workers: P530, P480, P950, P665, P880, P600
Arranged: P480, P530, P600, P665, P880, P950
Assigned numbers: 1, 2, 3, 4, 5, 6
$N = 6$ (even), so the median is the mean of the $\frac{6}{2} = 3$rd and $(\frac{6}{2} + 1) = 4$th values.
The 3rd value is P600 and the 4th value is P665.
$\frac{P600 + P665}{2} = P632.50$
Hence, the median daily wage of the six workers is P632.50.

Your Turn

Find the median of each data set.

1. Quiz scores: 15, 11, 18, 12, 11, 17, 21, 22, 17
   Arranged: 11, 11, 12, 15, 17, 17, 18, 21, 22
   Median: 17

2. Grade levels: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
   Arranged: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
   Median: 6.5

3. Heights (in cm): 140, 165, 155, 143, 142, 160, 159, 148
   Arranged: 140, 142, 143, 148, 155, 159, 160, 165
   Median: 151.5

4. Shoe sizes: 8.5, 7.0, 8.0, 9.0, 9.5, 8.5, 7.5, 8.0, 10.5, 9.5
   Arranged: 7.0, 7.5, 8.0, 8.0, 8.5, 8.5, 9.0, 9.5, 9.5, 10.5
   Median: 8.5

Mode of Ungrouped Data

• Denoted by $\hat{x}$ (x-hat).

• Most frequently occurring value in a set of data.

• Applicable for data in the nominal, ordinal, interval, or ratio levels of measurement.

• To find the mode, determine the value(s) that appear the most number of times.

  • One mode: unimodal distribution

  • Two modes: bimodal distribution

  • Three or more modes: multimodal distribution

  • All values appearing the same number of times: no mode

Example 4

a. Favorite colors: red, blue, green, green, blue, white, brown, black
Mode(s): green and blue; bimodal distribution
b. Religions of students: Baptist, Catholic, Baptist, Catholic, Catholic, INC, Catholic, Aglipayan, INC
Mode(s): Catholic; unimodal distribution
c. Quiz scores: 12, 11, 9, 15, 16, 13, 8, 4, 5, 17
Mode(s): The data has no mode.

Your Turn

Determine the mode. Then, describe it according to the number of modes.

1. Favorite flowers: rose, sampaguita, gumamela, sampaguita, rose, ilang-ilang, sampaguita
   Mode(s): sampaguita; unimodal

2. Exam grades: 80, 90, 75, 80, 90, 75, 78, 94
   Mode(s): 80, 90, 75; multimodal

3. Gender of participants: male, female, female, male, male, male
   Mode(s): male; unimodal

4. Rank of honor students: 1st, 3rd, 2nd, 1st, 2nd, 3rd
   Mode(s): 1st, 2nd, 3rd; multimodal

Summary of Properties and Uses of the Measures of Central Tendency

• Refer to the QR code for a tabulated summary.

Example 5

Determine which measure of central tendency is appropriate to use in each situation.
a. To determine if a value lies at the upper or lower 50% of the data set
The measure to use is the median because a data set must be separated into upper or lower half (50%).
b. Scores of a class with no extremely high nor low values
The measure to use is the mean because the data is under the interval/ratio level with no extremely high or low values.
c. Preference of customer on the type of burger patty
The measure to use is mode because the data is under the nominal level.

STOP AND THINK

• The mean of seven numbers is 65. If the mean of the first five numbers is 63, what is the mean of the last two numbers?

  • Let $x<em>1, x</em>2, …, x<em>7$ be the seven numbers. Then $\frac{x</em>1 + x<em>2 + … + x</em>7}{7} = 65$
    $x<em>1 + x</em>2 + … + x<em>7 = 455$ The mean of the first five numbers is 63: $\frac{x</em>1 + x<em>2 + x</em>3 + x<em>4 + x</em>5}{5} = 63$
    $x<em>1 + x</em>2 + x<em>3 + x</em>4 + x<em>5 = 315$ Then, the sum of the last two numbers is: $x</em>6 + x<em>7 = 455 - 315 = 140$ So the mean of the last two numbers is: $\frac{x</em>6 + x_7}{2} = \frac{140}{2} = 70$

• What is the mean, median, and mode of the first ninety-nine counting numbers?
  The first 99 counting numbers are 1, 2, 3, …, 99.
  Mean: $\frac{1 + 2 + 3 + … + 99}{99} = \frac{\frac{99 * 100}{2}}{99} = \frac{4950}{99} = 50$
  Median: Since there are 99 numbers, the median is the $\frac{99 + 1}{2} = 50$th number, which is 50.
  Mode: Since each number appears only once, there is no mode.

• The average of eight numbers is 250. Find the expression for the new average when two numbers 3x and 2x are added to it.
  Let $x<em>1, x</em>2, …, x<em>8$ be the eight numbers. Then, $\frac{x</em>1 + x<em>2 + … + x</em>8}{8} = 250$
  $x<em>1 + x</em>2 + … + x<em>8 = 2000$ If two numbers 3x and 2x are added, the new average is: $\frac{x</em>1 + x<em>2 + … + x</em>8 + 3x + 2x}{10} = \frac{2000 + 5x}{10} = 200 + \frac{x}{2}$

MATH FYI: The Weighted Mean

• Data has elements/values with greater weights than the rest. Examples:

  • Grades on subject areas (major subjects)

  • Price of gasoline (different unit prices)

• Formula: $\text{Weighted mean} = \frac{\sum wx}{\sum w}$

  • $w$: weight of each value or quantity $x$

Example 6

The following are grades of a student in his first year college subjects.

To get the weighted mean, multiply the grade by its corresponding weight. Then, add these products.

Since $\sum wx = 1129$ and $\sum w = 13$
$\text{Weighted Mean} = \frac{1129}{13} = 86.85$

MATH ENGAGE: Mean of Means

1. Prepare seven balls of the same size and color. Mark these balls with numbers 1 through 7. Put these balls inside a box.

2. Pick one ball, record the number on it, and then return it inside the box.

3. Pick another, record, and then return it again inside. Do the process until you have recorded four numbers.

4. Get the mean of your numbers.

5. Give the box of balls to four of your classmates. Let them do the process you did.

6. Compare the means you got. Now, get the mean of the five means. What value is it close to?

Interpret Data

Community Survey: Family size

Number of family members: 3, 4, 4, 4, 5, 6, 10

• Mean: $(36 ÷ 7) = 5.14$

• Mode: 4

• Best Measure: Mode or Median
  The mode and median both show that most families have 4 members.