Question

An Illinois state program evaluator is tasked with studying the intelligence of soon-to-graduate high school students in a number of Chicago-area high schools.

One of the specific questions that needs to be answered is, “How do the students of Collins High School, one of Chicago’s lowest-rated high schools in terms of academic achievement, fare in intelligence compared to students of Lincoln Park High School, one of Chicago’s highest-rated high schools in terms of academic achievement?”.

To conduct this study, the program evaluator administers the Wechsler Adult Intelligence Scale, 4th Edition (WAIS-IV) to one 12th grade class from each high school in the Chicago area (if you are interested in learning more about the WAIS-IV, click here).

The following table shows the WAIS-IV scores for student from Collins HS and Lincoln Park HS (note: data were fabricated for purposes of this excersize):

Collins HS		Lincoln Park HS
Student	WAIS-IV Score	Student	WAIS-IV Score
1	105	1	93
2	81	2	90
3	102	3	87
4	90	4	109
5	95	5	106
6	110	6	104
7	90	7	109
8	100	8	104
9	80	9	115
10	90	10	112
11	84	11	112
12	81	12	100
13	90	13	97
14	107	14	90
15	101	15	104
16	90	16	107
		17	101

First, complete the below grouped frequency table of WAIS-IV scores for each HS:

WAIS-IV Score	Collins HS ( f )	Lincoln Park HS ( f )
80-89
90-99
100-109
110-119

Compute the appropriate calculations to complete the following table :

MEASURE	Collins HS ( f )	Lincoln Park HS ( f )
Mean
Median
Mode
N
N-1
ΣX
(ΣX)2
ΣX2
S2X
SX
s2X
sX

What is the shape of the distribution of intelligence scores (normal, negatively skewed, positively skewed) for Collins HS? Explain how you arrived at your answer.

What is the shape of the distribution of intelligence scores (normal, negatively skewed, positively skewed) for Lincoln Park HS? Explain how you arrived at your answer.

Answer 1

Based on the given data, using excel,

The frequency table may be created as follows:

Thus, counting the colored cells would give the required frequencies:

Measures:

- Mean = Sum of the observation value / Total no. of observations

For Collins,

Mean =

= 93.5

Similarly for Lincoln Park, Mean = 102.4

- Median

It is the middle no. of the sorted series of observations.

For Collins, the sorted series is given by,

80,81,81,84,90,90,90,90,90,95,100,101,102,105,107,110.

Since it is an even series, median is the average of the two middle values i.e.

For Lincoln Park, since it is an odd series,

87,90,90,93,97,100,101,104,104,104,106,107,109,109,112,112,115,

Median is the middle value = 104.

- Mode

Mode is the most frequent observation in the series.

In case of Collins, Mode = 90

And for Lincoln Park, Mode = 104

- N = No. of observations in the data

For Collins, N= 16 and for  Lincoln Park, N = 17

-N-1 = 15 ...(For Collins)

= 16....(For Lincoln Park)

- ΣX denotes the sum of the observations.

Using the function 'SUM()' in excel,

ΣX = 1496 ...(For Collins)

= 1740....(For Lincoln Park)

- (ΣX)²   denotes the square of the sum of the observations

(ΣX)² = 2238016 ...(For Collins)

= 3027600....(For Lincoln Park)

- ΣX² is obtained by squaring each observation and summing the squares.

ΣX² = 141282 for Collins

= 179236 for Lincoln Park

- s²_x denotes the sample variance of the observations.

It is obtained by the formula

=

= 93.7

Similarly, for Lincoln Park, s²_x = 71.4

- s_x is the standard deviationof the observations, the square root of its variance, obtained as:

s_x = = 9.7.....for Collins

= 8.4 for Lincoln Park

Summarizing,

To determine the skewness of the data, we may use histogram:

From the histograms, we find that:

For Collins, Mean>Median. Also we find that a major bulk of the data lies towards the prior half of the graph.Hence, we may conclude that the data exhibits a slight positive skewness (i.e. longer right tail).

For Lincoln Park, Mean < Median.Also, we can observe a long left tail for the distribution.It implies that the data is negatively skewed.