MODULE 2

Mean, Median, Mode, Variance, Standard Deviation and Z-Scores

MODULE 2

Mean, Median, Mode, Variance, Standard Deviation and Z-Scores

A class of 8th graders at one school have the following test scores on a standardized test:

EXAMPLE

67 68 69 73 74 75 76 76 78 81 82 82

83 84 85 85 85 85 86 87 88 90 91 94

The **mean** is the arithmetic average of all the numbers; it is found using the following formula:

M = ΣX

N

M = Mean

Σ = Sum

X = Scores in the distribution of the variable X

N = Number of scores in the distribution

Using this formula, we can find the **Mean** for the above example:

1,944 / 24 = 81

**MEAN**

70% OF NEGLECT CASE

The **Mode** is the score that occurs most frequently.

In the case of the above example:

Mode = 85

**MODE**

The **Median** is the middle score of the group. To find the Median, complete the following:

Using this formula, we can find the **Median **for the above example:

(82 + 83) / 2 = 82.5

Empty text

- Line up the scores in numerical order.
- Count the number of scores, add 1, and divide by 2

**MEDIAN**

- In the case of an odd number of scores, such as 19, you add 1 (making 20), and divide by 2, meaning the 10th score is the Median.
- In the case of an even number of scores, such as 24, you add 1 (making 25), and divide by 2, meaning 12.5. Add the 12th and 13th scores and divide by 2 to find the Median

Charlie scores a 90 on the standardized test. Where does Charlie rank among those who have taken this test?

To know this, we need to convert Charlie’s score to a **Z-score**.

EXAMPLE

Next, we’ll discuss **variability**.

This is a numerical representation of how spread out the scores are around the Mean.

We’ll focus on two measurements, **Variance** and **Standard** **Deviation.**

To convert a raw score (90) to a Z-score: Subtract the mean from the raw score.Divide by the standard deviation.

To find **Variance, use the formula: **

SD² = Σ (X-M)²

N

- Find each score’s deviation score, but subtracting the mean from each score.
- Square each deviation score.
- Add all the square deviation scores together (commonly called
**sum of squared deviations**). - Find the Mean of the sum of squared deviations.

1256 / 24 = 52.333...

Using this formula, we can find the **Variance **for the above example:

**VARIANCE**

67

68

69

73

74

75

76

76

78

81

82

82

83

84

85

85

85

85

86

87

88

-14

-13

-12

-8

-7

-6

-5

-5

-3

0

1

1

2

3

4

4

4

4

5

6

7

196

169

144

64

49

36

25

25

9

0

1

1

4

9

16

16

16

16

25

36

49

If we take our example and arranged the scores into a table, this is how it looks:

X

X-M

X-M^2

Total

1,944

1.256

Mean = 81

Variance = 52.33333

SD = 7.234178

**Standard Deviation** is the positive square root of variance. It is the most widely used number when describing a spread of scores, and is used in future calculations.

√52.333... = 7.234

Using this formula, we can find the **Variance **for the above example:

**STANDARD DEVIATION**

SD = √SD²

**Z-scores** offer us a standardized way for comparing an individual score on a distribution of scores.

To convert a raw score (90) to a Z-score: 1) Subtract the mean from the raw score. 2) Divide by the standard deviation.

Z = ( 90 - 81) / 7.234 = 1.244

The raw score of 90 is 1.244 standard deviations away from the Mean of 81.

Using this formula, we can find the **Z-score **for the above example:

**Z - SCORES**

SD = √SD²

Z = X - M

SD

Looking at a Z-table, we can see that a Z-score of 1.244 is in the 89th percentile:

LOADING AWESOME

Charlie Z = 1.218

LOADING AWESOME

LOADING AWESOME